Googo- and googolple- (archived 3-part unfinished article)

alright guys, here’s another old article from my large number site. it’s about andre joyce’s googo- and googolple- system, and it was in fact the very first article on my site. for a while it lurked around as an unfinished 3-part page, but i have now finished a new article on that system. for history’s sake i will archive the old unfinished googo- and googolple- article on this blog.


Googologist Andre Joyce has made some odd observations about the googol and googolplex, and devised a googo- and googolple- number system. This page will showcase the two.

Googo-x is (2x)^x, where x is written in Roman numerals.
If you don’t remember, they’re:
I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1,000
Since L is 50, googol would evaluate to 100^50 = 10^100, which is correct.
Googolple-x needs some modification though, because Joyce somehow fucked up the definition (in fact, most of his googology is broken and needs fixes to be able to work right). The correct definition would be x^x^x^2, where x is in roman numerals. Googolplex here would be 10^10^10^2 = 10^10^100, which is once again correct.
Let’s examine googo- first:
Googo-1 = googoi (goo-goy) = 2^1 = 2, which is trivial even in everyday life.
Googo-2 is differently named. One would think it’s googoii, but that can’t be pronounced any differently from googoi. So Joyce allows us to use ij in place of ii Therefore it’s googoij (goo-goyj), or 4^2 = 16. In day to day life it’s a sizable number (for example, having 16 kids is outrageous), but easy for us to comprehend, and trivial on the larger scale.
Since Joyce says we should use “ox” in place of “iii”, googo-3 should be called googoox, but that’s too odd, and confusable with “x” (10). I instead suggest “iji” instead, so googo-3 is googoiji (goo-goy-jee). Googoiji evaluates to 6^3 = 216, which is sizable in the everyday world and generally impressive, but still palpable to the mind.
Googo-4 is called “googoiv” (goo-goyv). It evaluates to 8^4 = 4,096, around the limits of what our minds can visualize.
Googo-5 is called googov (goo-gawv). It’s equal to 10^5, or exactly 100,000 – called lakh in the Indian number system.
Googo-6 is called googovi (goo-gaw-vye) = 12^6 = 2,985,984, or about three million. It’s approximately the population of Missisippi.
Googo-7 = googovij (goo-gawv-ij) = 14^7 = 105,413,504, a bit less than the population of Mexico.
Googo-8 = googoviji (goo-guh-vee-jee) = 16^8 = 4,294,967,296, around the population of Asia
Googo-9 = googoix (goo-goyx) = 18^9 = 189,359,290,368. Comparable to the number of stars in the Milky Way Galaxy.
Googo-10 = googox (goo-gawx) = 10,240,000,000,000, or 10 trillion. Comparable to the number of human cells in your body, and slightly more than the googologically important megafuga-three (7,625,597,484,987).
Googo-11 = googoxi (goo-gawx-ee) = 584,318,301,411,328, or 584 trillion. This many miles is around 10 light years, or the distance to the star Sirius.

Googo-12 = googoxij (goo-gawx-ij) = 36,520,347,436,056,576, or 36 quadrillion. This many miles is around 6,000 light years, or the distance to the Perseus Spiral Arm in the Milky Way Galaxy.

Googo-13 = googoxiji (goo-gawx-ee-jee) = 2,481,152,873,203,736,576, or 2.5 quintillion. This many miles is around 425,000 light years, or four times the diameter of the Milky Way.

Googo-14 = googoxiv (goo-gawx-iv) = 182,059,119,829,942,534,144, or 182 quintillion. This many miles is around 31 million light years, or the distance to the Sombrero Galaxy.

Googo-15 is a bit trickier, as googoxv is hard to pronounce. Fortunately, Joyce recently addd “vy” as in “ivy” in place of v. “googoxvy“, therefore, is pronounced “googox-vee”. It’s approximately 1.435*10^22, or 14 sextillion. At this scale it’s harder to come up with good and precise examples for the magnitude of these numbers.

Googo-16 = googoxvi (goo-gawx-vye) ~ 1.209*10^24, or 1.2 septillion.

Googo-17 = googoxvij (goo-gawx-vij) ~ 1.084*10^26, or 108 septillion.

Googo-18 = googoxviji (goo-gawx-vee-jee) ~ 1.031*10^28. This number is bigger than the diameter of the observable universe in meters!

Googo-19 = googoxix (goo-gawx-ix) ~ 1.037*10^30.

Googo-20 = googoxex (Joyce allows us to use “ex” instead of “x” to make it pronounceable) ~ 1.100*10^32.

We can continue withgoogoxexigoogoxexijgoogoxexijigoogoxexivgoogoxexvygoogoxexvigoogoxexvijgoogoxexviji, and googoxexix, but by now we can go at a faster rate and take jumps of tens.

Googo-30 = googoxexex ~ 2.210*10^53. That’s roughly the amount of atoms in Jupiter.

Googo-40 = googoxel ~ 1.329*10^76. Googoxeliji is the first googo- number more than the number of atoms in the observable universe, but still less that the number of small volumes in large spaces (like Planck volumes in the universe), and much less than many combinatorical numbers such as the odds of a monkey typing out Hamlet on its first try.

Googo-50 = the famed googol = 10^100. It’s often regarded as impractically large, but that isn’t true – it’s still less than, say, the number of Planck volumes in the universe. It’s also not too much less than (for instance) the number of ways to arrange a 6x6x6 Rubik’s cube (that’s around 10^116), so if you happen to get your hands on one, you have a real example of a number bigger than a googol RIGHT ON YOUR FINGERTIPS!

Googo-60 = googolex, a number Joyce writes on his website. It’s around 10^124.
Googo-70 = googolexex ~ 10^150. That’s almost as much as 4^^3 = 4^4^4 ~ 10^153.
Googo-80 = googolexexex ~ 10^176.
Googo-90 = googoxcy, since “cy” (as in Nancy) is offered as an alternative to c. In pronunciation, it could be confused with googoxi, so I’ll make it pronounced “goo-gawks-kee” instead. This number is around 10^203, or slightly more than a gargoogol (googol^2 = 10^200). In reality it’s a thousand times larger, but you get the point. It’s more than the number of Planck volumes in the observable universe!
Now for googo-s of 3-digit numbers:
Googo-100 = googoc (goo-gawk) ~ 10^230. This one seems to be brought up particularly often, probably due to its simple name. It’s equal to bingol (2^100) times gargoogol (10^200).
With that we can give a whole table of the first 100 googo- numbers:
1 2
2 16
3 216
4 4096
5 100,000
6 2.98598E+6
7 1.05413E+8
8 4.29497E+9
9 1.98359E+11
10 1.024E+13
11 5.84318E+14
12 3.65203E+16
13 2.48115E+18
14 1.82059E+20
15 1.43489E+22
16 1.20892E+24
17 1.08428E+26
18 1.03144E+28
19 1.03726E+30
20 1.09951E+32
21 1.22528E+34
22 1.43205E+36
23 1.75158E+38
24 2.23763E+40
25 2.98023E+42
26 4.13129E+44
27 5.95157E+46
28 8.89741E+48
29 1.37851E+51
30 2.21074E+53
31 3.66557E+55
32 6.27709E+57
33 1.10905E+60
34 2.01977E+62
35 3.78818E+64
36 7.31086E+66
37 1.45068E+69
38 2.95744E+71
39 6.18997E+73
40 1.32923E+76
41 2.92662E+78
42 6.6028E+80
43 1.52557E+83
44 3.60776E+85
45 8.72796E+87
46 2.15893E+90
47 5.45767E+92
48 1.40935E+95
49 3.716E+97
50 1.E+100
51 2.74542E+102
52 7.68659E+104
53 2.19387E+107
54 6.38091E+109
55 1.89059E+112
56 5.70439E+114
57 1.75217E+117
58 5.47726E+119
59 1.74196E+122
60 5.63475E+124
61 1.85332E+127
62 6.19647E+129
63 2.10545E+132
64 7.26839E+134
65 2.54869E+137
66 9.07568E+139
67 3.28111E+142
68 1.20405E+145
69 4.48391E+147
70 1.69419E+150
71 6.4934E+152
72 2.52406E+155
73 9.94852E+157
74 3.97527E+160
75 1.61007E+163
76 6.60865E+165
77 2.7485E+168
78 1.15803E+171
79 4.94208E+173
80 2.13599E+176
81 9.34794E+178
82 4.14186E+181
83 1.85768E+184
84 8.43295E+186
85 3.87399E+189
86 1.80072E+192
87 8.46807E+194
88 4.02823E+197
89 1.93812E+200
90 9.43029E+202
91 4.63976E+205
92 2.30802E+208
93 1.16066E+211
94 5.89981E+213
95 3.03104E+216
96 1.57369E+219
97 8.25596E+221
98 4.37617E+224
99 2.34344E+227
100 1.26765E+230
In exponential functions, the ratio between f(x) and f(x-1) is constant. But the ratio between googo-x and googo-(x-1) increases, albeit slowly. Therefore googo-x grows slightly faster than exponentiation.
Googo-101 = googoci (goo-gawk-ee) ~ 10^232
Googo-110 = googocex (goo-goh-sex) ~ 10^258
Googo-150 = googocel (goo-goh-sel) ~ 10^372
Googo-200 = googocc (goo-gawch) ~ 10^520
Googo-201 = googocci (goo-gawch-ee). Its name is reminiscent of Italian, and it’s about 10^523.
Googo-300 = googoccc. That’s pronounced goo-goshk. It’s about 10^833.
Googo-400 = googocdy (goo-gawk-dee) ~ 10^1161. That’s more than Joyce’s great googol (10^1000)! Joyce offers “dy” (as in lady) as an alternative to “d”.
Googo-500 = googod. That’s exactly 10^1500, and it has an awesome name, doesn’t it?
Googo-600 = googodcy (goo-god-see) ~ 10^1847
Googo-666 = googobeast. Just kidding. It’s called googodecelexvi (goo-goh-dek-uh-lex-vye) ~ 10^2081.
Googo-700 = googodccy (goo-god-chee) ~ 10^2202
Googo-800 = googodcccy (goo-god-shkee) ~ 10^2563
Googo-900 = googocem (goo-goh-sem) ~ 10^2930
And finally:
Googo-1000 = googom ~ 10^3,301
Googo-1337 = googoleet = 10^133,337. Just kidding. It’s actually googomcccexexexvij (goo-gawm-shkex-ex-ex-vij) ~ 10^4582.
Googo-2000 = googomem ~ 10^7204
Googo-3000 = googomemem ~ 10^11,334
We can go further only with a more confusing -bar system Joyce devises, but I won’t discuss that. Instead, in an upcoming article I’ll discus Joyce’s crazy extension, producing insane googologisms like geiggeim.
Remember that googolple-x is x^x^x^2, where x is in Roman numerals. From there we can devise some whole new numbers, far far larger than the googo- group.
Let’s start:
Googolple-1 = googolplei (goo-guhl-play, not to be confused with Google Play) = 1^1^2 = 1^1 = 1. Heh, pathetically trivial.
Googolple-2 = googolpleij (goo-guhl-playj) = 2^2^4 = 2^16 = 65,536. A good jump from before, but not quite … scary, to say the least. It’s between googoiv and googov, closer to googov.
Googolple-3 (googolpleiji [goo-guhl-play-jee]) is much larger. It’s equal to 3^3^9 = 3^19,683 = 

Wow! That’s impressive,  but it’s still possible to write in full, even by hand – its expansion is above. This number is 9,392 digits long, placing it between googomem and googomemem, and around the breaking point of the googo- system without the bar stuff.

How about the NEXT number?

Googolple-4 = googolpleiv (goo-guhl-playv) = 4^4^16 = 4^4,294,967,296.

We can approximate this in scientific notation as follows: = 10^log(4^4,294,967,296) = 10^(4,294,967,296*log(4)) ~ 10^(2,585,827,972.983642077204) = 10^2,585,827,972 * 10^0.983642077204 ~ 9.63035*10^2,585,827,972. We’ll need to use googo- on numbers in the billions to get to a number this big! It’s too large to directly store on this website, but it’s possible to put as a downloadable text file. Even then, it’s ridiculously big for a text file, and would take up a rather enormous portion of a hard drive.

From there on we can now that it’s around 2.6 billion digits long, making it comparable to Jonathan Bowers’ nanillion (10^3,000,000,003, the billionth -illion). It begins 9630350133920400475534…..

This number can still digitally be written out in full, but it would take up gigabytes of space. The last digits are harder to compute, but still possible. Here’s what we’ll do for the last two:

4^1 ends 4
4^2 ends 16
4^3 ends 64
We can continue, knowing that the 4*(the last two digits as a 2-digit number) will be the last two digits of (4*the whole number)
4^4 ends 56
4^5 ends 24
4^6 ends 96
4^7 ends 84
4^8 ends 36
4^9 ends 44
4^10 ends 76
4^11 ends 04
Then it starts over, and cycles at a sequence of 10 terms. From here on out we can determine that the last 2 digits of 4^x = f(x mod 10), where f(n) is the last 2 digits of 4^n, which can be looked up above.
4,294,967,296 mod 10 is equal to 6, so we can say this number ends …..96. More digits are possible, but it’s a little trickier from here on. I might do it sometime, but for know, here’s the beginning and end of googolpleiv: 96303……(2.6 billion digits)…..96.
Up next is googolplev, which is equal to 5^5^25, which is around 5^(2.98*10^17). In terms of 10 it’s around 10^(2.08*10^17), or around E17#2 (that’s 10^10^17) in Hyper-E Notation. In Hyper-E, Ea#b = 10^10^10…..^10^b, with a 10s. The number is around 200 quadrillion digits long. It’s possible to compute both the leading and ending digits easily, but let’s do that later. Storing this number is possible on a supercomputer with around an exabyte (exabyte is 1 million terabytes) of space or around a million terabyte files, but it’s hard and time-consuming to do so.
Googolplevi is next in line – it’s equal to 6^6^36 ~ 10^(8.01*10^27) ~ E28#2 – that’s 8 octillion digits long! It can’t be stored on all the data in the world (somewhere in the zettabytes, a zettabyte is 10^21 (sextillion) bytes or a billion terabytes), but it probably will be possible in the future … but it’s safe to say that it’s quite a while before we are able write this monster in full. It would take up over a brontabyte of space to store in full.
The seventh member of the series is the googolplevij, equal to 7^7^49 ~ 10^(2.7*10^41) ~ E41#2. Writing this number is near impossible, but it’s hard to say whether it’s “possible” or “impossible”. It will require technology over a QUINTILLION (that’s a million million million, or a billion billion) TIMES more powerful than what we have today (keep in mind that a quintillion is HUGE), and let’s just say that.
Number the eighth is googolpleviji = 8^8^64 ~ 10^10^58 = E58#2. The number of digits is a billion times the number of atoms on Earth!!! It’s impossible to write out, as we’ll need a billion Earths at LEAST, but definitely more because most atoms on planets are in the inaccessible core. The number of atoms on Earth can be considered a limit of “writable”, but it’s still kind of arbitrary.
What’s next? The googolpleix (goo-guhl-playx) = 9^9^81 ~ 10^10^78 = E78#2 – the number of digits is approximately the number of atoms in the observable universe, and a hard limit of “writable” – even numbers at this scale are guaranteed to be unwritable.
And now for number ten – the legendary googolplex = 10^10^100. It’s a well-established statement that it’s impossible to write a googolplex, but unlike the notion that “there isn’t a googol of anything”, this one is true. But it can still represent real-world things – for instance Sbiis Saibian’s promaxima is around 10^10^343, and it’s an estimate of the number of possible histories of the universe, start to finish, assuming Planck units aren’t meaningful.
These first ten terms alone are a pretty interesting sequence. But what unfathomable stuff lies ahead?
Googolplexi is next in line. It’s equal to 11^11^121 ~ E126#2.
Now let’s skip to googolplexvy (goo-guhl-plex-vee) = 15^15^225 ~ E265#2 – the number of digits this has is more than the number of PLANCK VOLUMES in the observable universe!
Googolple-20 = googolplexex ~ E520#2. As you can see the growth rate of googolple- is faster than hyper-exponential! Hyper-exponential growth is when the number of digits grows exponentially. But it’s still much much slower than hyper-hyper-exponential, where the number of digits increases hyper-exponentially.
Googolple-30 = googolplexexex ~ E1329#2.
Googolple-40 = googolplexel ~ E2563#2
Googolple-50 = googolplel ~ E4247#2
Googolple-60 = googolplelex ~ E6401#2
Googolple-100 = googolplec = 10^(2*10^20,000) – even the exponent now is becoming very, very large.
Googolple-200 = googolplecc (goo-guhl-plech) ~ E92,041#2 ~ E5#3. This number has nearly agoogolgong (10^100,000) digits!!!
Googolple-500 = googolpled ~ E674,742#2, between E5#3 and E6#3, closer to the latter. Writing the number is impossible, and now even the exponent takes up megabytes of space.
Googolple-1000 = googolplem = 1,000^1,000^1,000,000 = 10^(3*10^3,000,000),, between E6#3 and E7#3. Interestingly, this number (hyper-exponentially) is very close to Bowers’ megillion, equal to 10^(3*10^3,000,003+3)
Since the googolple- system is now near its breaking point, here’s a table showing the number of digits in each of the first 100 googolple- numbers:
1 1
2 5
3 9,392
4 2.58583E+9
5 2.08309E+17
6 8.02616E+27
7 2.17125E+41
8 5.66879E+57
9 1.8763E+77
10 1.E+100
11 1.06201E+126
12 2.72391E+155
13 2.01043E+188
14 5.01565E+224
15 4.90879E+264
16 2.16463E+308
17 4.89555E+355
18 6.41251E+406
19 5.45551E+461
20 3.35958E+520
21 1.65957E+583
22 7.25215E+649
23 3.0768E+720
24 1.38551E+795
25 7.21084E+873
26 4.70658E+956
27 4.16797E+1043
28 5.3999E+1134
29 1.10092E+1230
30 3.78904E+1329
31 2.35583E+1433
32 2.82571E+1541
33 6.96754E+1653
34 3.75607E+1770
35 4.69881E+1891
36 1.4457E+2017
37 1.15751E+2147
38 2.54746E+2281
39 1.62527E+2420
40 3.16712E+2563
41 1.98234E+2711
42 4.18861E+2863
43 3.13491E+3020
44 8.71242E+3181
45 9.41136E+3347
46 4.13369E+3518
47 7.71512E+3693
48 6.38841E+3873
49 2.44711E+4058
50 4.52028E+4247
51 4.19088E+4441
52 2.02981E+4640
53 5.33931E+4843
54 7.92156E+5051
55 6.88236E+5264
56 3.6319E+5482
57 1.20749E+5705
58 2.61835E+5932
59 3.8362E+6164
60 3.93012E+6401
61 2.91081E+6643
62 1.61056E+6890
63 6.88122E+7141
64 2.34325E+7398
65 6.56447E+7659
66 1.56052E+7926
67 3.24437E+8197
68 6.08464E+8473
69 1.05937E+8755
70 1.76313E+9041
71 2.88807E+9332
72 4.78992E+9628
73 8.27211E+9929
74 1.52903E+10236
75 3.10779E+10547
76 7.13901E+10863
77 1.9011E+11185
78 6.03099E+11511
79 2.33782E+11843
80 1.13599E+12180
81 7.09369E+12521
82 5.83438E+12868
83 6.48633E+13220
84 9.96906E+13577
85 2.17224E+13940
86 6.86866E+14307
87 3.22675E+14680
88 2.30471E+15058
89 2.55862E+15441
90 4.52104E+15829
91 1.2993E+16223
92 6.2089E+16621
93 5.04631E+17025
94 7.12066E+17434
95 1.78366E+17849
96 8.10317E+18268
97 6.80852E+18693
98 1.0816E+19124
99 3.31545E+19559
100 2.E+20000
The number of digits of the exponents increases at a rate we will call R. R grows, but slowly. It can be thought of as the difference between the number of digits of the exponent of googolple-x and the number of digits of the exponent of googolple-(x-1). In a hyper-exponential function (like 10^10^x), the ratio between the logarithm of f(x) and f(x-1) is constant, but for googolple-x it increases slowly. Therefore it grows slightly faster than hyper-exponential functions.
In conclusion, the googo- and googolple- systems are pretty cool, but they don’t match with the goal of modern googology, which is to invent, analze, and understand new notations instead of just devising whimsical naming systems. But then again, at the time of Joyce’s writing googology was at its early larval stages, so of course there was some odd experimenting. This may also tell us why Joyce’s notation is broken; he didn’t find googology to be serious business and just some playful experimenting, not bothering to check how big his numbers are. Then again, Bowers’ work (which existed at the time) already was like modern googology, except for many larger numbers having a definition left for googologists to fully formalize and understand themselves.

What’s next? Joyce continued his googo- system with a crazy vowel and letter repeating scheme – it’s actually kind of cool.


Note: Andre Joyce changed his extended googo- system to make it much less powerful – however, I prefer the more powerful old one, which I will discuss here.

Remember the googo- system? Well, Joyce decided to continue it with a crazy scheme. Here it is:

A googol type number is written like so, according to Joyce: (modified from the g function to standard mathematical/googological forms)

a, b, c, d are each letters or combinations of letters
a and c are vowels determined by variables A and C, respectively:
0 o (pronunciation depends on context)
1 oo (as in food)
2 ee (as in feet)
3 or (as in born)
4 ie (i as in fine)
5 i (as in kid)
6 e (as in leg)
7 ei (a as in game)

(the pronunciations were by me)

b is a sequence of g’s, the number of g’s is a variable B minus 1
d is the variable D written in Roman numerals (modifications allowed for pronounceability are ij for ii, iji for iii, ex for x, el for l, ec for c if it isn’t preceded by another c, ed for d, and em for m)
Gabcd = (B{C}D){A}D
(where {x} is ^^^…^^^ with x ^s, and {0} is *)
First let’s check that our rules properly evaluate googol:
In googol, we need to find A, B, C, and D.
A = 1 (oo)
B = 2 (one g)
C = 0 (o)
D = 50 (L is 50 in Roman numerals)
Then plug in the values:
= 10^100, which is correct.
Now let’s work out some examples, to better comprehend this crazy system.
1. Examples with “o” as vowels

Numbers of this type are of the form gobod, where b and d are described above. Let’s work out some examples:

First, the smallest example, gogoi (goh-goy). First we determine A, B, C, and D:

A = 0 (o)
B = 2 (one g)
C = 0 (o)
D = 1 (i in roman numerals)

Then we plug the values in:

= 2*1*1
= 2. Trivial value.

Next, we try gogoij (goh-goyj):
A = 0
B = 2
C = 0
D = 2
= 2*2*2
= 8. Somewhat bigger.
Third number in the sequence, gogoiji:
A = 0
B = 2
C = 0
D = 3
= 2*3*3
= 18. Still bigger, but so far growing quite slowly.
Here we can notice some patterns: only D in the sequence is changing. We can devise a definition for the gogo- prefix:
gogo-x (where x is written in Roman numerals) = (2{0}x){0}x = 2*x*x = 2x2 (that’s two times the square of x)

With that, it’s trivial to continue with some values: (for pronunciation, refer to part 1 of googo- and googolple-)

x number name value
4 gogoiv 32
5 gogov 50
6 gogovi 72
7 gogovij 98
8 gogoviji 128
9 gogoix 162
10 gogox 200
11 gogoxi 242
15 gogoxvy 450
20 gogoxex 800
30 gogoxexex 1,800
40 gogoxel 3,200
50 gogol* 5,000
51 gogoli 5,202
52 gogolij 5,409
60 gogolex 7,200
70 gogolexex 9,800
80 gogolexexex 12,800
90 gogoxcy (goh-gox-kee) 16,200
100 gogoc 20,000

Hmmm. These first 100 aren’t that impressive, even by everyday standards. The gogo- function only has modest quadratic growth – it’s literally a quadratic function. But let’s continue.

*Not to be confused with googol, or Jonathan Bowers’ goggol (that’s 10{6}100). Though that could be a continuation of the giggol, gaggol, geegol, gigol, goggol, gagol series (giggol, gaggol, geegol, etc are 10{x}100 where x is 2, 3, 4, etc, so giggol is 10{2}100 = 10^^100, gaggol is 10^^^100, geegol is 10^^^^100, etc). None of these numbers are to be confused with any of my googo- numbers (like googoxex vs gogoxex).

101 gogoci 20,402
102 gogocij 20,808
110 gogocex 24,200
120 gogocexex 28,800
150 gogocel 51,200
200 gogocc (goh-goch) 80,000
300 gogoccc (goh-goshk) 180,000
400 gogocdy 320,000
500 gogod 500,000
600 gogodec 720,000
700 gogodecc (goh-goh-dech) 1,008,200
800 gogodeccc (goh-goh-deshk) 1,280,000
900 gogocem 1,620,000
1,000 gogom 2,000,000
1,100 gogomec 2,420,000
1,500 gogomed 4,500,000
2,000 gogomem 8,000,000
3,000 gogomemem 18,000,000

We can see that this system (not counting the -bar stuff) reaches its breaking point in the ten millions order of magnitude – the gogo- prefix is relatively weak, even compared to the humble googo- prefix, whose breaking point is over a millillion (10^3003, the thousandth -illion).

But we can go further with examples only using “o” as a vowel by repeating the “g”. Let’s try some examples, with two g’s:

First: goggoi (gog-oy). First we find A, B, C, and D:

A = 0 (o)
B = 3 (two g’s)
C = 0 (o)
D = 1 (i in roman numerals)

Then we calculate:

= 3*1*1
= 3. Once again trivial.

The second term, goggoij (gog-oyj) is similar, but d is 2 instead of 1. So we evaluate:

= 3*2*2
= 12
Right here we can conclude that a formula for goggo-x is 3x^2 (that’s three times the square of x),where x is written in Roman numerals (same variations allowed as previous).

We continue with a table, this time shorter than previous:

x number name value
3 goggoiji 27
4 goggoiv 48
5 goggov 75
10 goggox 300
20 goggoxex 1,200
50 goggol* 7,500
100 goggoc 30,000
200 goggocc 120,000
500 goggod 750,000
1,000 goggom 3,000,000
2,000 goggomem 12,000,000
3,000 goggomemem 27,000,000

*Goggol is also the name of a Bowersism equal to 10^^^^^^100 (that’s 6 ^s), which is FAR FAR FAR UNFATHOMABLY larger. I shall call this “goggol” the Joycian Goggol, as it’s created with a Joycian system. But this number is so small that it’s full decimal expansion (7,500) is easy to remember, so why bother naming it?

Well, this certainly grows faster than previous, but literally only 50% faster – not even twice as fast. Let’s try numbers with more g’s:

(gogg…ggo-x with y g’s in a row is equal to (y+1)*x^2)

gogggol (go-guh-guh-guhl) = 4*50^2 = 10,000
gogggom (go-guh-guh-gom) = 4*1000^2 = 4,000,000
gogggomemem (go-guh-guh-goh-mem-em) = 4*3000^2 = 36,000,000
goggggom (go-guh-guh-guh-gom) = 5*1000^2 = 5,000,000
gogggggom (go-guh-guh-guh-guh-gom) = 6*1000^2 = 6,000,000

This whole system with only o’s as vowels only reaches the ten millions order of magnitude here – with more g’s, the g’s get hard to keep track of, and are no longer worthy of continuing. Even then, with larger (20-100) amounts of g’s, the system barely reaches a billion, due to the innocent quadratic growth rates (with cubic growth rates overall).  Overall, they can’t go further than my quadratic number group in my Pointless Gigantic List of Numbers. So now let’s explore the possibilities of adding the “oo” vowel – these numbers will probably reach exponential/hyper-exponential levels.

2. Examples with “oo” as vowels
Here we can reach new possibilities with “oo” – here we’ll try numbers that start with goo- instead of go-.
We’ll start with the googo- prefix, which we know is (2x)^x, and give an example:
A = 1 (oo)
B = 2 (one g)
C = 0 (o)
D = i (1 in roman numerals)
googoi = (2{0}1){1}1 = (2*1)^1 = 2.
googo-x = (2{0}x){1}x = (2*x)^x = (2x)^x, where x is written in roman numerals
Here’s a table of some more numbers to recap from part 1:
x number name value
2 googoij 16
3 googoiji 216
4 googoiv 4,096
5 googov 100,000
6 googovi 2,985,984
7 googovij 105,413,504

Wow! The seventh member of the sequence already transcends ANY number we’ve covered so far. Let’s continue:

8 googoviji 4.3E9
9 googoix 1.9E11
10 googox 1.0E13
11 googoxi 5.8E14
12 googoxij 3.7E16
13 googoxiji 2.5E18
14 googoxiv 1.8E20
15 googoxvy 1.4E22
20 googoxex 1.1E32
30 googoxexex 2.2E53
40 googoxel 1.3E76
50 googol 1E100 (exactly equal)
51 googoli 3E102
As you can see, the 51st entry even transcends the googol – oh wait, googol is part of the googo- naming system. Let’s finish up: (keep in mind that E(n) means 10^n, so E100 = 10^100 = googol)
60 googolex (1) E124
70 googolexex E150
80 googolexexex E176
90 googoxcy E203
100 googoc E230
101 googoci E232
110 googocex E258
150 googocel E372
200 googocc E520
201 googocci E523
299 googoccexecix (googoccic) (2) E830
300 googoccc E833
400 googocdy E1161
500 googod E1500 (exactly equal)
600 googodec E1847
700 googodecc E2202
800 googodeccc E2563
900 googocem E2930
990 googocemexec (googoxem) (2) E3264
1000 googom E3301
1003 googomiji (googolmox) (3) E3312
2000 googomem (googomump) (4) E7204
3000 googomemem E11,334 (5)
(1) Italic names are defined by Joyce himself.
(2) Joyce offers alternative number naming (example: ic here means 99, because c is 100 and i is 1, 100 – 1 = 99) to make numbers easier to pronounce (I don’t offer this, for consistency with Roman numerals’ sake).
(3) Joyce offers “ox” as an alternative to “iii” just as I offer the less confusing “iji”. By Joyce’s system googo-1003 should be googomox, but he makes a mistake and calls it googolmox.
(4) Joyce offers both “em” (which I use) and “ump” as alternatives to “m”.
(5) Joyce also defines a few -bar numbers, but like I said, they’re not of interest here.
So as you can see, the googo- system easily crushes all numbers with only “o” as vowels – but it’s only the start of more cool numbers. It’s breaking point is just over a great great googol (10^10,000).
Behold the googgo- prefix. Let’s calculate how much googgo-x evaluates to (we’ll leave D as the variable x):
A = 1 (oo)
B = 3 (two g’s)
C = 0 (o)
… and plug it in …
… and now we’ll test some values.
x number name value
1 googgoi (goo-guh-goy)* (3*1)^1 = 3
2 googgoij (goo-guh-goyj) (3*2)^2 = 36
3 googgoiji (goo-guh-goyj-ee) (3*3)^3 = 729

* Sbiis Saibian wrote in his article on Joyce’s googology (which serves mostly to show how broken it is) that he doesn’t know how to distinguish googol, googgol, and googgool in speech, and doubts Joyce cares. Here is my answer – just pronounce each repeated g individually as “guh”.

The first three members of the googo- sequence (3, 36, 729) are already noticeably larger than their googo- counterparts (2, 16, 216). Let’s continue:

4 googgoiv 20,736 – that’s 5x larger than googoiv.
5 googgov 759,375 – 7.5x larger than googov
6 googgovi 34,012,224 – 11x larger than googovi
7 googgovij 1.8E9 – 18x larger than googovij
8 googgoviji 1.1E11 – 44x larger than googoviji
9 googgoix 7.6E12 = 7,625,597,484,987 – that’s exactly equal to the important megafugathree, which is 3^3^3 = 3^27. Megafugathree shows up a lot when working with powers of 3.
10 googgox 5.9E14 – that’s almost 600x larger than googox.

As you can see, these numbers grow notably faster than googo-. Let’s continue:

11 googgoxi 5.1E16
12 googgoxij 4.7E18
13 googgoxiji 4.8E20
14 googgoxiv 5.3E22
15 googgoxvy 6.3E24
20 googgoxex 3.7E35
30 googgoxexex 4.2E58
40 googgoxel 1.5E83
47 googgoxelvij 3E101 – this is the first larger than a googol.
50 googgol (goo-guh-guhl) 6E108 – defined by Joyce himself.
60 googgolex 2E135
70 googgolexex 4E162
80 googgolexexex 3E190
90 googgoxec 7E218
100 googgoc 5E247
200 googgocc 4E555
300 googgoccc 2E866
500 googgod E1588
1000 googgom E3477
2000 googgomem E7556
3000 googgomemem E11,863
Hmm. Let’s compare this with googo-, with the largest example on both lists.
Googgo-3000 is around E11,863, while googo-3000 is around E11,334. This makes the first 10^529 times larger than the second. Unfortunately, we’re at an exponential scale with these numbers, so we’re looking more at the difference between the exponents. Therefore, the difference is no longer very large at this scale.
So let’s continue with a smattering of the many possible triple-G numbers. First we define the googggo- prefix…
googggo-x = (4x)^x, where x is written in Roman numerals
… and then we define numbers …
x number name value
1 googggoi (goo-guh-guh-goy) 4
2 googggoij 64
3 googggoiji 1,728
4 googggoiv 65,536
5 googggov 32,000,000
6 googggovi 1.9E8
7 googggovij 1.3E10
8 googggoviji 1.1E12
9 googggoix 1.0E14
10 googggox 1.0E16
15 googggoxvy 4.7E26
20 googggoxex 1.2E38
25 googggoxexv E50 (exactly equal)
30 googggoxexex 2.4E62
45 googggoxelvy 3E101 (first term more than a googol)
50 googggol 1E115
75 googggolexexv 6E185
100 googggoc 2E260
200 googggocc 4E580
250 googggoccel E750 (exactly equal)
500 googggod E1651
1000 googggom E3602
3000 googggomemem E12,238
We can see that this function (on a logarithmic scale) grows only slightly faster than the previous. Let’s give a smattering of the ones with quadruple g’s:
number name value
googgggoi (5*1)^1 = 5
googgggoij (5*2)^2 = 100
googgggox (5*10)^10 ~ 9.8E16
googgggoxex E40 (exactly equal)
googgggol 8E119
googgggoc 8E269
googgggocc E600 (exactly equal)
googgggom E3699
googgggomem E8000 (exactly equal)
googgggomemem E12,528

Like you probably expected, this one grows ever so slightly faster than the previous. Now let’s try a few with five or more g’s:

googggggol (6*50)^50 ~ 7E123
googggggoc 7E277
googggggom E2778
googgggggom E3845
googggggggom E3903

Since it’s hard to keep track of all the g’s now, we can safely say that numbers with “oo” then “o” as vowels are at a breaking point. These numbers reach their breaking point between a great great googol (10^10,000) and a googolgong (10^100,000), and have exponential growth rates overall. They can be upper-bounded by a googolgong.

So what’s next? Let’s try having “oo” as both the first and second vowel, and see what possibilities that opens.
First, as usual, let’s try the smallest example: googooi (goo-goo-ee).
A = 1 (oo)
B = 2 (one g)
C = 1 (oo)
D = i (1 in roman numerals)
= (2^1)^1
= 2

The overall formula for googoo-x is (2^x)^x, which simplifies to 2^x^2 (note that this is 2^(x^2), NOT (2^x)^2). Let’s try some examples:
x number name value
2 googooij 2^2^2 = 2^4 = 16. Interestingly, this is exactly equal to googoij.
3 googooiji 2^3^2 = 2^9 = 512
4 googooiv 2^4^2 = 2^16 = 65,536
5 googoov 33,554,432
6 googoovi 6.9E10
7 googoovij 5.6E15
8 googooviji 1.9E19
9 googooix 2.4E24
10 googoox* 1.3E30
11 googooxi 2.7E36
12 googooxij 2.2E43
13 googooxiji 7.5E50
14 googooxiv 1.0E59
15 googooxvy 5.4E67
16 googooxvi 1.2E77
17 googooxvij 9.9E86
18 googooxviji 3.4E97
19 googooxix 5E108

*This is why I don’t like using “ox” in place of “iii” – the third member of the googo- series (googo-3) would be called googoox with Joyce’s system, but this is already an interpretation of googoox. Joyce’s googology is ambiguity galore!

Impressive so far! The 19th term already passes the googol, while the previous sequences could only pass it after forty-something terms. The exponent itself is growing at a quadratic rate, making this sequence faster than exponential growth, but still much much slower than hyper-exponential, where the exponent itself increases hyper-exponentially. Let’s see what happens when we continue:
20 googooxex 3E120
30 googooxexex 9E270
40 googooxel 4E481
50 googool 4E752 – that’s more than googol to the seventh!
60 googoolex E1084 – passed a great googol!
70 googoolexex E1475
80 googoolexexex E1927
90 googooxcy E2438
100 googooc E3010
150 googoocel E6773
200 googoocc E12,041
300 googooccc E27,093 – larger than anything we’ve seen so far!
400 googocdy E48,615
500 googood E75,258 – good number, isn’t it?
1000 googoom E301,030 – bigger than a googolgong!
2000 googoomem E1,204,199 – even passes a millionplex (also called four-ex-great googol)
3000 googoomemem E2,709,270
This system goes well beyond anything we’ve previously seen! It passes the googolgong, and even the millionplex – the smallest major googologism above it is Jonathan Bowers’ micrillion (the millionth -illion, equal to 10^3,000,003), and it’s a fairly strong bound, on a logarithmic scale.
Now what will googgoo- (goo-guh-goo) bring? We can calculate the formula of googgoo-x as 3^x^2, and calculate from there:

x number name value
1 googgooi (goo-guh-goo-ee) 3
2 googgooij 81
3 googgooiji 19,683 – sizable difference so far (googooiji, for comparison, is 512)
4 googgooiv 43,046,721
5 googgoov 8.5E11
6 googgoovi 1.5E17
7 googgoovij 2.4E23
8 googgooviji 3.4E30
9 googgooix 4.4E38
10 googgoox 5.2E47
15 googgooxvy 2E107 – first one bigger than googol
20 googgooxex 7E190
30 googgooxexex 3E429
40 googgooxel 3E763
50 googgool E1193 – defined by Joyce himself
60 googgoolex E1718
100 googgooc E4771
200 googgoocc E19,085
300 googgooccc E42,941
500 googgood E119,280
1000 googgoom E477,121
2000 googgoomem E1,908,485
3000 googgoomemem E4,294,091

These numbers are pretty cool – the exponents become over 1.5x as large as their googoo- counterparts. This passes a micrillion but is easily beaten by Bowers’ nanillion (10^3,000,000,003, the billionth -illion). Now let’s give a few numbers with triple g’s:
x number name value
1 googggooi (4^1)^1 = 4
2 googggooij (4^2)^2 = 256
3 googggooiji 262,144
4 googggooiv 4.3E9 – interestingly, exactly equal to googoviji from part 1 and earlier here
5 googggoov 1.1E15
6 googggoovi 4.7E21
10 googggoox 1.6E60
13 googggooxiji E102 – first one larger than googol
15 googggooxvy E135
20 googggooxex  E241
50 googggool E1505
100 googggooc E6021
500 googggood E150,515
1000 googggoom E602,060
3000 googggoomemem E5,418,540

… and then with more g’s …

googgggool E1748
googgggooc E6990
googgggoom E698,970
googggggoom E778,151
googgggggoom E845,098
googgggggoomemem E7,605,882
Not that the googgggggoom (with six g’s in a row) is almost equal to googoom CUBED.

But all these numbers are still bounded by E10,000,000, and can also be upper-bounded by a billionplex (10^10^9).

There’s still more though (though they won’t be nearly as big as the previous numbers). These unusual constructions will use “o” as a vowel and then “oo”.
First, an example:
A = 0 (o)
B = 2 (one g)
C = 1 (oo)
D = 50 (L = 50 in Roman numerals)
= (2{1}50}{0}50
= (2^50)*50
~ 5.6E16, or 56 quadrillion. Quite modest compared to many of the previous numbers, which were millions of digits long, but still very large in everyday terms. It’s even far far less than a googol.
The gogoo- prefix can be defined as x*2^x, where x is written in Roman numerals. Let’s try some examples:
x number name value
1 gogooi (goh-goo-ee) 2 – note how the frst member of a sequence with one g in a row is 2, with two g’s in a row it’s 3, with three g’s it’s 4, etc.
2 gogooij 8
3 gogooiji 24
4 gogooiv 64 – so far not very impressive.
5 gogoov 160
6 gogoovi 384
7 gogoovij 896
8 gogooviji 2048 – also the name of an infamous game
9 gogooix 4608
10 gogoox 10,240
15 gogooxvy 491,520
20 gogooxex 20,971,520
30 gogooxexex 3.2E10
40 gogooxel 4.4E13
50 gogool 5.6E16
75 gogoolexexv 2.8E24
100 gogooc 1.3E32
150 gogoocel 2.1E47
200 gogoocc 3.2E62
324 gogoocccexexiv (goh-goosh-kex-ex-iv) 1.1E100 – first to pass a googol
500 gogood E153
1000 gogoom E304
2000 gogoomem E605
3000 gogoomemem E906

One thing to point out: gogoo-x is EXACTLY equal to f2(x) in the fast growing hierarchy. (if you don’t know, f0(x) = x+1, f1(x) = f0(f0(f0…..(f0(x))….)) nested x times = 2x, f2(x) = f1(f1(f1…..(f1(x))….)) = x*2^x, f3(x) = f2(f2(f2…..(f2(x))….)) nested x times, etc. It has been extended to far far bigger proportions involving bigger infinite ordinals, might discuss that in other pages.)

This system’s numbers start out rather unimpressive, but they do go faster: they eventually reach the astronomical range, and then something beyond that! It’s certainly far far faster than numbers with only “o” as vowels, but far weaker than googoo-, googgoo-, and the like and even slower than googo-. This one doesn’t make it to a great googol. Nevertheless, let’s try numbers with more g’s here.

First off:
goggoo-x (pronounced gog-oo-[x] where x depends on the number) = x*3^x
gogggoo-x (pronounced go-guh-guh-goo-[x] where x depends on the number) = x*4^x
goggggoo-x (pronounced go-guh-guh-guh-goo-[x] where x depends on the number) = x*5^x
Now for the double-g numbers:
x number name value
1 goggooi 3
2 goggooij 18
3 goggooiji 81
4 goggooiv 324
5 goggoov 1215
6 goggoovi 4374 – note that gogoovi is only 384
7 goggoovij 15,309
8 goggooviji 52,488 – not to be confused with 524,288, which is the 19th power of 2
9 goggooix 177,147
10 goggoox 590,490
15 goggooxvy 215,233,605
20 goggooxex 7.0E10
30 goggooxexex 6.1E15
40 goggooxel 4.9E20
50 goggool (gog-ool) 3.6E25 – compare to gogool, which is around 5.6E16
75 goggoolexexv 4.6E37
100 goggooc 5.2E49
200 goggoocc 5.3E97
205 goggooccvy (go-gooch-vee) 1.3E100 – first to pass a googol. This sequence passes a googol 36% earlier than the previous.
300 goggooccc E146
500 goggood E241
1000 goggoom E480
2000 goggoomem E958
3000 goggoomemem E1435
This sequence passes the great googol, but not by too far and doesn’t reach a millillion.
Let’s just look at the 3000th members of the next four sequences:
gogggoomemem E1810
goggggoomemem E2100
gogggggoomemem E2338
goggggggoomemem E2538

None of them, not even the last, reach a millillion – it takes nine g’s in a row to make it to that number, and by then the g’s get hard to manage – let’s just say the go-x-oo-y family can be bounded by a millillion.

2.5. Hierarchy of numbers so far

Now’s a good time to put up a hierarchy of numbers so far.

There are two levels so far: those with o as the highest vowel and those with oo as the highest vowel. The higher “ee”, “or”, etc. levels will come later.

The first level has one family (the o-o family, like gogol and goggol), and the second has three (from lowest to highest, o-oo (like gogool), oo-o (like googol), and oo-oo (like googool)).

And each family is further divided into multiple series, depending on how much the g is repeated. For example, the oo-o family has the googo-, googgo-, googggo-, etc. series.

Each series is a group of numbers. For example, the googo- series starts with googoi, googoij, googoiji, etc. and also includes higher members like googox, googol, and googoc.

1. cap is an upper bound for all numbers in a family/level.

2. First number in parentheses after a series name is the formula for numbers in the

3. Second number in parentheses after a series name is the value of the 50th member (for example, googol for the googo- series)


So here’s a visualization:
o level – cap: E8
    o-o family – cap: E8
        gogo- series (2x^2 / 5000)
        goggo- series (3x^2 / 7500)
        gogggo- series (4x^2 / 10,000)
        goggggo- series (5x^2 / 12,500)
oo level – cap: E7#2 = E10,000,000
    o-oo family – cap: E3000
        gogoo- series (x*2^x / 5.6E16)
        goggoo- series (x*3^x / 3.5E25)
        gogggoo- series (x*4^x / 6.3E31)
        goggggoo- series (x*5^x / 4.4E36)
    oo-o family – cap: E100,000
        googo- series ((2x)^x / E100*)
        googgo- series ((3x)^x / E108)
        googggo- series ((4x)^x / E115)
        googgggo- series ((5x)^x / E120)
    oo-oo family – cap: E7#2 = E10,000,000
        googo- series (2^x^2 / E753)
        googgo- series  (3^x^2 / E1193)
        googggo- series  (4^x^2 / E1505)
        googgggo- series  (5^x^2 / E1747)
* exact value
This should give us a quick overview of how powerful each system is, as well as its behavior.
So what’s next? The more powerful vowels of course. These will have chaotic growth rates. The tetrational ones (with ee) will be harder to work with, the pentational ones (with or) harder still, and hexation and beyond will be still even harder.
3. Numbers with “ee” as vowels

In the googo- naming system, “ee” is the third vowel – it’s equivalent to tetration (a^^b).

Let’s try an example to see its power:

geegol (also the name of a Bowersism equal to 10^^^^100, but this is a different Joycian geegol)
A = 2 (ee)
B = 2 (one g)
C = 0 (o)
D = 50 (L)
= (2{0}50){2}50
= (2*50)^^50
= 100^^50 = 100^100^100……^100 with 50 100’s = HUGE!
100^^50 is an EXTREMELY UNFATHOMABLE number – equal to 100^100^100^100…….^100 with 50 100’s. It’s larger than ANY number to appear in science at all – it’s unfathomably larger than a googolplex, -duplex, -triplex, -quadriplex, -quintiplex, and the decker (10^^10)! In terms of 10s, it can be approximated as 10^10^10^……..10^200 with 49 10’s, or just E200#49 – placing it just slightly larger than a penantalogue (10^^50), in tetrational terms.
But seriously, where the hell did that come from? We need to find out using examples…..lots of examples.
Let’s start small, and introduce the hierarchy of prefixes (families sorted from weakest to strongest):
ee level
    o-ee family
        gogee- series
        goggee- series
    oo-ee family
        googee- series
        googgee- series
    ee-o family
        geego- series
        geeggo- series
    ee-oo family
        geegoo- series
        geeggoo- series
    ee-ee family
        geegee- series
        geeggee- series
…and then we’ll go through each one.
First the gogee- series ((2^^x)*x):
gogeei (goh-gee-ih) = (2^^1)*1 = 2*1 = 2 – note that the first member of any such series is ALWAYS degenerate, and equal to c in the formula.
gogeeij (goh-gee-ij) = (2^^2)*2 = (2^2)*2 = 4*2 = 8. Very pathetic so far.
gogeeiji (goh-gee-ij-ee) = (2^^3)*3 = (2^2^2)*3 = (2^4)*3 = 16*3 = 48
gogeeiv (goh-gee-iv) = (2^^4)*4 = (2^2^2^2)*4 = (2^16)*4 = 262,144 – certainly far larger, but certainly doesn’t strike fear into us.
gogeev = (2^^5)*5 ~ 1.0E19,729 – HOLY FUCKING SHIT THIS IS SO MUCH BIGGER THAN ANYTHING PREVIOUSLY ENCOUNTERED. But it’s still a real-world level number, and a number still less than the higher numbers in the oo level (like googoom). It’s also smaller than a googolgong, and smaller than fairly well-known numbers in mathematics, like the largest known primes (around 10^10^7). It’s also feasibly writable, even on this website without having to scroll too long. Nevertheless, this is still a very huge number, and let’s see how huge the next one is.
gogeevi = (2^^6)*6 ~ E19,728#2 – WHAT? THAT’S INSANE! This number is very unfathomably large – its number of digits is approximately the previous number. It even passes the googolplex, and any number we’ve encountered so far here (in parts 1 and 2) EXCEPT for the 100th and higher members of the googolple- series (googolplec and above). It’s completely impossible to write out, even if you put a digit in every Planck volume in the known universe (which is impossible in its own right). But this is still less than a googolduplex, a tetralogue (10^10^10^10), or even a googolplexigong (10^10^100,000, the gong version of a googolplex).
gogeevij = (2^^7)*7 ~ E19,728#3 – the exponent is approximately the previous number! This passes the googolduplex.
gogeeviji = (2^^8)*8 ~ E19,728#4 – passes the googoltriplex
gogeeviji = (2^^9)*9 ~ E19,728#5
gogeeix = (2^^10)*10 ~ E19,728#6
gogeex = (2^^11)*!1 ~ E19,728#7
Staggering, right? Yep – this easily crushes anything we’ve encountered so far. It’s also a great example of tetrational growth rate. What is tetrational growth? The growth rate of a function like this – the size of the power tower increases by one in each step, usually not until after a few steps. Here’s another few functions with tetrational growth:
The -logue series (L(x) = 10^^x): L(1) = 10, L(2) = 10^10 = dialogue, L(3) = 10^10^10 = E1#3 = trialogue, L(4) = 10^10^10^10 = E1#4 = tetralogue, L(5) = E1#5 = pentalogue, etc.
The megafuga- series (M(x) = x^^x): M(1) = 1, M(2) = 4, M(3) ~ E13, M(4) ~ E154#2, M(5) ~ E2184#3, M(6) ~ E36,305#4, M(7) ~ E695,974#5 ~ E6#6, M(8) ~ E7#7, M(9) ~ E9#8, M(10) = E10#9 = E1#10.
For a more complicated example, try the Catalan-Mersenne Sequence – it is the series 2, M2, MM2, MMM2, MMMM2, etc. (M[x] = (2^x)-1, so M1 = 1, M2 = 3, M3 = 7, M4 = 15, etc. Some of those numbers (3, 7, 31, 127…..) are prime and called Mersenne primes.)
Let’s try the first few terms:
CM(1) = 2, CM(2) = M2 = 2^2-1 = 4-1 = 3, CM(3) = MM2 = 2^M2-1 = 2^3-1 = 8-1 = 7, CM(4) = 2^M3-1 = 2^7-1 = 127, CM(5) = 2^M4-1 = 2^127-1 ~ 1.7E38, CM(6) ~ E38#2, CM(7) ~ E38#3, CM(8) ~ E38#4, etc.
The first five here are all known to be prime, but the sixth is too large for any known primality test, but is still conjectured to be prime.
What do these sequences have in common? The power tower height increases by around one in every step, after a few terms. The -logue series grows like this right away, megafuga- series grows like this from the third term, and the Catalan-Mersenne sequence from the fifth term. This is such a new type of growth rate – one that will be familiarized as we go through these later series.
Without further ado let’s finish the rest of the gogee- series:
x number name value
20 gogeexex E19,278#17
30 gogeexexex E19,278#27
40 gogeexel E19,278#37
50 gogeel E19,278#47
100 gogeec E19,278#97
200 gogeecc E19,278#197
500 gogeed E19,278#497
1000 gogeem E19,278#997
2000 gogeemem E19,278#1997
3000 gogeememem E19,278#2997
Pretty insane – this passes Bowers’ giggol (10^^100) and even my great giggol, analogous to the great googol, equal to 10^^1000. Its numbers EASILY dwarf anything from parts one and two, even the largest members of the googolple- series!
But this series is by necessity less than the goggee- (gog-ee) series (3^^x)*x…..but by how much? That’s what we’ll find out next.
First things first – goggeei (gog-ee-ih) = (3^^1)*1 = 3 – once again degenerate.
goggeeij = (3^^2)*2 = 3^3*2 = 27*2 = 54. A sizeable jump, but barely clears my secondary number range (7 up to about 50) in my number list.
goggeeiji = (3^^3)*3 = 3^3^3*3 = 3^27*3 = 3^27*3^1 = 3^(27+1) = 3^28 = 22,876,792,454,961. Exactly three times megafugathree.
goggeeiv = (3^^4)*4 ~ 10^(3.6*10^12), between E12#2 and E13#2 – this is a few trillion digits long, and when written out, would take terabytes of space.
goggeev = (3^^5)*4 ~ E13#3 – this is impossible to write.
goggeevi ~ E13#4
goggeex ~ E13#8
goggeel ~ E13#48
goggeec ~ E13#98
goggeed ~ E13#498
goggeem ~ E13#998
goggeemem ~ E13#2998
The gogee- sequence is always around one power tower height behind the goggee- sequence – both grow at tetrational rates. Gogggeee-, goggggeee-, etc. behave similar, each ending slightly ahead of the previous. Their formulas are (4^^x)*4, (5^^x)*5, etc.
But what about the googee- family? Each series is by necessity faster than its gogee- counterpart, but by how much? We’ll find out here. First we make the formulas:
googee-x = (2^^x)^x
googgee-x = (3^^x)^x
googggee-x = (4^^x)^x
googgggee-x = (5^^x)^x
Then we calculate the values:
googeei = 2
googeeij = (2^^2)^2 = 4^2 = 16
googeeiji = (2^^3)^3 = 16^2 = 4096
googeeiv ~ 1.84E19 – so far satisfyingly faster than gogee-, and decently astronomical compared to gogeeiv, which is only in the tertiary range
googeev ~ E98,641 –  has about twice as many digits as gogeev.
googeevi ~ E19,729#2 – what? That’s almost the same approximation given for gogeevi. This is because powers only multiply the exponent, which loses its effectiveness once the exponent becomes exponential in size.
googeevij ~ E19,728#3 – ditto.
googeeviji ~ E19,728#4
googeeix ~ E19,728#5
googeex ~ E19,728#6
and the next googee- numbers are indistinguishable at a tetrational scale from their gogee- counterparts. Ditto for the googgee- series – see for yourself.
So the oo-ee family and the o-ee family, really, are on par with growth rates.
But what about some different families? First let’s try the ee-o family:
We’ll start off with what geego-x means: it means (2x)^^x, where x is in Roman numerals.
In any case, it’s trivial to continue:
geegoi = 2^^1 = 2
geegoij = 4^^2 = 256
geegoiji = 6^^3 ~ 10^(2.65*10^36,304) – a big jump, but still writable in full (see full decimal expansion).
geegoiv = 8^^4 ~ 10^10^15,131,337 – another huge jump, and impossible to write in full
geegov = 10^^5 = 10^10^10^10^10 – exactly equal to a pentalogue
geegovi = 12^^6 ~ E13#5
geegovij = 14^^7 ~ E16#6
geegoviji = 16^^8 ~ E19#7
geegoix = 18^^9 ~ E23#8
geegox = 20^^10 ~ E26#9
geegoxex = 40^^20 ~ E64#19
geegol = 100^^50 ~ E200#49
geegoc = 200^^100 ~ E460#99
As you can see, this sequence grows faster than the previous ones…..but it’s only a faster version of tetrational growth. Nevertheless, it’s still a little faster than typical tetrational growth, similar in rate to the megafuga- prefix.

this is the end of the googo- and googolple- article series. as i said it’s mostly archived for history’s sake – the new content is currently under construction and will be much better.

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