(***Original Text by Aarex Tiaokhiao***)
***The following is the original version of Aarex Tiaokhiao's Nested Cascading-E Notation with BEAF numbers. Here he includes many numbers.
Nested Cascading-E Notation with BEAF numbers
As a Cascading-E Notation, he reaches Great and Terrible Tethrathoth.
So here a extension of Cascading-E Notation:
A new rules is Uses a #^…^(#...#) = #^…(#^…^(#...)).
Nested Cascading-E Notation (E^…)
Other name “Cascading Hyper-E Notation”
E(a)b = a^b
# 1 = #
# n#m = # (# n#m-1)
# #1m = # #m
# n #l m = # n #l-1 n...n #l-1 n (m n's)
# n #^# m = # n #...# m (n #'s)
# n #^^@ m = n #^#...#^# (@ #^#'s)
# n #^...^@ m = # n #^...(#^...^@-1) m
# n @/# m = # n @-1 m
... = #^^# = #^# --> X
#^#*# --> X+1
#^#*## --> X+2
#^#*### --> X+3
...
#^#*#^# --> 2X
#^#*#^#*# --> 2X+1
#^#*#^#*## --> 2X+2
...
#^#*#^#*#^# --> 3X
...
#^#*#^#*#^#* ... *#^#*#^# w/n #^#s --> nX
I defined reverse-cascading symbol: */* --> *-*
Next we have #^## --> XX = X2. So the #^## has the same power as planar arrays. Continuing on we have:
#^## --> X2
#^##*# --> X2+1
#^##*## --> X2+2
...
#^##*#^# --> X2+X
#^##*#^#*# --> X2+X+1
...
#^##*#^#*#^# --> X2+2X
#^##*#^#*#^#*#^# --> X2+3X
...
#^##*#^## --> 2X2
#^##*#^##*#^## --> 3X2+X
...
#^### --> X3
#^#### --> X4
...
#^^## = #^#^# --> XX
Here we observe an interesting property of the comparison. The number of hashes in the exponent is also the number of dimensions of the comparable array. We can say if we have #^$(#) where $(#) is some expression composed of hyperions, replace all of these with X and we can say:
#^$(#) --> $(X)
Hence...
#^^### = #^#^#^# --> X^X^X
#^^#### = #^#^#^#^# --> X^X^X^X
#^^##### = #^#^#^#^#^# --> X^X^X^X^X
...
This notation reaches a #^^#^#, so here more:
#^^#^# = #^#...#^# (X ^'s) --> X^^X
#^(#^^(#^#/#)*#) --> X^(X^^(X-1)+1)
#^^(#^#*#) --> X^^(X+1)
#^^(#^#*##) --> X^^(X+2)
#^^(#^#*###) --> X^^(X+3)
#^^(#^#*#^#) --> X^^2X
#^^(#^#*#^#*#^#) --> X^^3X
#^^(#^##) --> X^^(X^2)
#^^(#^###) --> X^^(X^3)
#^^(#^#^#) --> X^^(X^X)
#^^(#^#^#^#) --> X^^(X^X^X)
#^^#^^#^# --> X^^X^^X
#^^#^^#^^#^# --> X^^X^^X^^X
How about pentation?
#^^^#^# --> X^^^X
Note: #^^^#^# = #^^#^^...#^^#^^#^# (X-1 ^^'s)
And hexation, heptation, ...?
#^^^^#^# --> X^^^^X, #^^^^#^# = #^^^#^^^...#^^^#^^^#^# (X-1 ^^^'s)
#^^^^^#^# --> X^^^^^X, #^^^^^#^# = #^^^^#^^^^...#^^^^#^^^^#^# (X-1 ^^^^'s)
In FGH, this growth rate is fφ(ω,0)(n) > E(n)n#^...^#^#n /w n+1 ^'s.
As Extensible-E System, Nested Cascading-E Notation are fourth term.
As Hyper-E Notation is first (E#), Extended Hyper-E Notation (xE#) is second and Cascading-E Notation (E^) are third.
Formly New E#^...
Get ready to reaching #^...^#^#...
Tethrathoth Group (Expanded)
Now revisit of tethrathoth:
Tethrathoth = E100#^^#^#100
Did you know: #^^# in Extended Cascading-E Notation is equal to #^^#^#
Grand Tethrathoth = E100#^^#^#100#2
Grantethrathoth = E100#^^#^#100#100
Greatethrathoth = E100#^^#^#100#100#100
Gigantethrathoth = E100#^^#^#100##4
Gorgetethrathoth = E100#^^#^#100##5
Gultethrathoth = E100#^^#^#100##6
Gasptethrathoth = E100#^^#^#100##7
Ginortethrathoth = E100#^^#^#100##8
There a E100#^^#^#100##100 exist, so
gugold ~ tethrathoth =
Gugoltethrathoth = E100#^^#^#100##100
Greetagoltethrathoth = E100#^^#^#100##100#100
Gugolthratethrathoth = E100#^^#^#100##100##100
throogol ~ tethrathoth =
Throotethrathoth = E100#^^#^#100###100
Tristo-Throotethrathoth = E100#^^#^#100###100
teroogol ~ tethrathoth =
Terootethrathoth = E100#^^#^#100####100
Petootethrathoth = E100#^^#^#100#####100
Ectootethrathoth = E100#^^#^#100######100
There a Throotethrathoth series, so here a 100th member:
Godgahtethrathoth = E100#^^#^#100#^#100
Wait, there more!
Grandgahtethrathoth = E100#^^#^#100#^#100#100
Gugoldgahtethrathoth = E100#^^#^#100#^#100##100
Graatagoldgahtethrathoth = E100#^^#^#100#^#100##100#100
Gugolthragahtethrathoth = E100#^^#^#100#^#100##100##100
Teroogahtethrathoth = E100#^^#^#100#^#100###100
Gotrigahtethrathoth = E100#^^#^#100#^#100#^#100
Gotergahtethrathoth = E100#^^#^#100#^#100#^#100#^#100
We define a grid, throo, teroo, ...
Godgridgahtethrathoth = E100#^^#^#100#^#*#100
Godthroogahtethrathoth = E100#^^#^#100#^#*##100
Godteroogahtethrathoth = E100#^^#^#100#^#*###100
Deutero-godgahtethrathoth = E100#^^#^#100#^#*#^#100
Trito-godgahtethrathoth = E100#^^#^#100#^#*#^#*#^#100
Gridgahtethrathoth = E100#^^#^#100#^##100
Gridtrigahtethrathoth = E100#^^#^#100#^##100#^##100
Gridtergahtethrathoth = E100#^^#^#100#^##100#^##100#^#100
Gridgodgahtethrathoth = E100#^^#^#100#^##*#100
Gridgrandgahtethrathoth = E100#^^#^#100#^##*##100
Gridgugoldgahtethrathoth = E100#^^#^#100#^##*#^#100
Gridthroogahtethrathoth = E100#^^#^#100#^##*#^#*#^#100
Gridugahtethrathoth = E100#^^#^#100#^##*#^##100
Kubikahtethrathoth = E100#^^#^#100#^###100
Godgatethrathoth = E100#^^#^#100#^#^#100
Godtothrathoth = E100#^^#^#100#^#^#^#100
We define a tetrithrathoth:
Tetrithrathoth = E100#^^#^#100#^^#^#100
Teterthrathoth = E100#^^#^#100#^^#^#100#^^#^#100
Tetppegathrathoth = E100#^^#^#100#^^#^#100#^^#^#100#^^#^#100
We define a tetgoldthrathoth:
Tetgoldthrathoth = E100#^^#^#*#100
Tetgridthrathoth = E100#^^#^#*##100
Tethroothrathoth = E100#^^#^#*###100
Teteroothrathoth = E100#^^#^#*####100
Tetpetoothrathoth = E100#^^#^#*#####100
Tetectoothrathoth = E100#^^#^#*######100
Tetzettoothrathoth = E100#^^#^#*#######100
Tetyottothrathoth = E100#^^#^#*########100
We define a deutero-tethrathoth and trito-tethrathoth:
Deutero-tethrathoth = E100#^^#^#*#^^#^#100
Trito-tethrathoth = E100#^^#^#*#^^#^#*#^^#^#100
This is end of expanded tethrathoth group.
Tethrathor Group
Okay, we imagined hecato-tethrathoth.
This is Tethrator = E100#^(#^^(#^#/#)*#)100
At short one, @ is #^^(#^^#/#)*#, so tethrator = E100#^@100.
Wait there more!
Grand Tethrator = E100#^@100#2
Grantethrator = E100#^@100#100
Greatethrator = E100#^@100#100#100
Gugoltethrator = E100#^@100##100
Throothrator = E100#^@100###100
We define a tetrithrator = E100#^@100#^@100, teterthrator = E100#^@*#4 and tetppegathrator = E100#^@*#5.
We define a tetgoldthrator = E100#^@*#100 and tetgridthrator = E100#^@*##100.
We define deutero-tetgoldthrator = E100#^@*#^@100 and trito-todathoth = E100#^@*#^@*#^@100.
Tetgridthrathor Group
Here a tetgridthrathor. Its a E100#^(#^^###...###*##)100
There more:
Grand Tetgridthrathor = E100#^(#^^(#^^#/#)*##)100#2
Grantetgridthrathor = E100#^(#^^(#^^#/#)*##)100#100
Tetrigridthrathor = E100#^(#^^(#^^#/#)*##)*#3
Tetergridthrator = E100#^(#^^(#^^#/#)*##)*#4
Tetppegagridthrator = E100#^(#^^(#^^#/#)*##)*#5
Deutero-tetgridthrathor = E100#^(#^^(#^^#/#)*##)*#^(#^^(#^^#/#)*##)100
Trito-tetgridthrathor = E100#^(#^^(#^^#/#)*##)*#^(#^^(#^^#/#)*##)*#^(#^^(#^^#/#)*##)100
Tethroothrathor Group
A tethroothrathor is equal to E100#^(#^^(#^#/#)*###)100
Teteroothrathor is equal to E100#^(#^^(#^#/#)*####)100
Tetpetoothrathor is equal to E100#^(#^^(#^#/#)*#####)100
Tetectoothrathor is equal to E100#^(#^^(#^#/#)*######)100
Tetzettoothrathor is equal to E100#^(#^^(#^#/#)*#######)100
Tetyottoothrathor is equal to E100#^(#^^(#^#/#)*########)100
Using a terrible tethrathoth series:
Tethrathoth = E100#^^#^#100
Terrible Tethrathoth = E100(#^^#^#)^^#^#100
Terrible Terrible Tethrathoth = E100((#^^#^#)^^#^#)^^#^#100
...
Tethrathoth ba'al = E100(...#^^#^#...)^^#^#100 /w 100 ^^#^#'s
...
Great and Terrible Tethrathoth = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s
...
Great and Terrible Tethrathoth ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s
...
Great and Terrible Tethrathoth super ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s
...
Great and Terrible Tethrathoth duper ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s
...
Great and Terrible Tethrathoth tripler ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s
...
Great and Terrible Tethrathoth teripler ba'al = E100(...#^^#^#...)^^#^#100 /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w E100(...#^^#^#...)^^#^#100 ^^#^#'s /w 100 ^^#^#'s
Tethrathordeus = E100#^(#^^(#^#/#))100
Tethrathortruce = E100#^(#^^(#^#/#)*#^^(#^#/#)*#^^(#^#/#))100
Tethrathorquad = E100#^(#^^(#^#/#)*#^^(#^#/#)*#^^(#^#/#)*#^^(#^#/#))100
Todatbiath Group
Todatbiath = E100#^#^(#^^###...###*#)100 \w 98 #'s
Trandatbiath = E100#^#^(#^^###...###*##)100 \w 98 #'s
Tugolthratbiath = E100#^#^(#^^###...###*#^#)100 \w 98 #'s
Tugoltriatbiath = E100#^#^(#^^###...###*#^##)100 \w 98 #'s
Throogolthratbiath = E100#^#^(#^^###...###*#^###...###)100 \w 98 #'s
Tris-throogolthratbiath = E100#^#^(#^^###...###*#^###...###*#^###...###)100 \w 98 #'s
Tessi-throogolthratbiath = E100#^#^(#^^###...###*#^###...###*#^###...###*#^###...###)100 \w 98 #'s
Todatriath Group and Beyond
Todatriath = E100#^#^#^(#^^###...###*#)100 \w 97 #'s
Trandatriath = E100#^#^#^(#^^###...###*##)100 \w 97 #'s
Tugolthratriath = E100#^#^#^(#^^###...###*#^#)100 \w 97 #'s
Throogolthratriath = E100#^#^#^(#^^###...###*#^###...###)100 \w 97 #'s
Tristo-throogolthratriath = E100#^#^#^(#^^###...###*#^###...###*#^###...###)100 \w 97 #'s
Teristo-throogolthratriath = E100#^#^#^(#^^###...###*#^###...###*#^###...###*#^###...###)100 \w 97 #'s
Todatetrath = E100#^#^#^#^(#^^###...###*#)100 \w 96 #'s
Trandatetrath = E100#^#^#^#^(#^^###...###*##)100 \w 96 #'s
Tulthratetrath = E100#^#^#^#^(#^^###...###*#^#)100 \w 96 #'s
Throogolthratetrath = E100#^#^#^#^(#^^###...###*#^###...###)100 \w 96 #'s
Tristo-throogolthratetrath = E100#^#^#^#^(#^^###...###*#^###...###*#^###...###)100 \w 96 #'s
Teristo-throogolthratetrath = E100#^#^#^#^(#^^###...###*#^###...###*#^###...###*#^###...###)100 \w 96 #'s
Beyond Tessi-throogolthratetrath
Gridthomegath = E100#^#...#^#^##100 \w 100 #'s
Throomegath = E100#^#...#^#^###100 \w 100 #'s
Tethomkillath = E100#^#...#^#^#^#100 \w 101 #'s
Gridthomkillath = E100#^#...#^#^#^##100 \w 101 #'s
Throomkillath = E100#^#...#^#^#^###100 \w 101 #'s
Tethomath = E100#^#...#^#^#^#^#100 \w 102 #'s
Tethomgigath = E100#^#...#^#^#^#^#^#100 \w 103 #'s
Tethomegator = E100#^#...#^#...#^#100 \w 200 #'s
Tethomkillator = E100#^#...#^#100 \w 201 #'s
Tethomator = E100#^#...#^#100 \w 202 #'s
Tethomegatohol = E100#^#...#^#...#^#100 \w 300 #'s
Tethomegatethol = E100#^#...#^#...#^#100 \w 400 #'s
Tethomegathousath = E100#^#...#^#...#^#100 \w 10000 #'s
Tethomegongath = E100#^#...#^#...#^#100 \w 1000000 #'s
Tethomegingath = E100#^#...#^#...#^#100 \w 100000000 #'s
Tethomegoath = E100#^#...#^#...#^#100 \w 100^100 #'s
Tethomegiath = E100#^#...#^#...#^#100 \w 100^100^100 #'s
Tethomegaath = E100#^#...#^#...#^#100 \w 100^100^100^100 #'s
Tethrathothatria Group
Ok the 1st group is godgahlah group and 2nd group is tethrathoth group.
This a 3rd group:
Tethrathothatria = E100#^^#^^#^#100
Tethomegoathatria = E100#^^#^^#^#^#100
Tethomegiathatria = E100#^^#^^#^#^#^#100
Tethomegaathatria = E100#^^#^^#^#^#^#^#100
Tethrathothatetra Group to Tethrathothaocta Group
For 4th group, we put -atetra at a end:
Tethrathothatetra = E100#^^#^^#^^#^#100
Tethomegoathatetra = E100#^^#^^#^^#^#^#100
Tethomegiathatetra = E100#^^#^^#^^#^#^#^#100
Tethomegaathatetra = E100#^^#^^#^^#^#^#^#^#100
For 5th group, we put -apenta at a end:
Tethrathothapenta = E100#^^#^^#^^#^^#^#100
Tethomegoathapenta = E100#^^#^^#^^#^^#^#^#100
Tethomegiathapenta = E100#^^#^^#^^#^^#^#^#^#100
Tethomegaathapenta = E100#^^#^^#^^#^^#^#^#^#^#100
For 6th group, we put -ahexa at a end:
Tethrathothahexa = E100#^^#^^#^^#^^#^^#^#100
Tethomegoathahexa = E100#^^#^^#^^#^^#^^#^#^#100
Tethomegiathahexa = E100#^^#^^#^^#^^#^^#^#^#^#100
Tethomegaathahexa = E100#^^#^^#^^#^^#^^#^#^#^#^#100
For 7th group, we put -ahepta at a end:
Tethrathothahepta = E100#^^#^^#^^#^^#^^#^^#^#100
Tethomegoathahepta = E100#^^#^^#^^#^^#^^#^^#^#^#100
Tethomegiathahepta = E100#^^#^^#^^#^^#^^#^^#^#^#^#100
Tethomegaathahepta = E100#^^#^^#^^#^^#^^#^^#^#^#^#^#100
And for 8th group, we put -aocta at a end:
Tethrathothaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#100
Tethomegoathaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#^#100
Tethomegiathaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#^#^#100
Tethomegaathaocta = E100#^^#^^#^^#^^#^^#^^#^^#^#^#^#^#100
Penthrathoth Group
Ok, let godgahlah for stage 1, tethrathoth for stage 2, tethrathothatria for stage 3, tethrathothatetra for stage 4,...
Let number is stage 100, penthrathoth which equal to E100#^^^#^#100 and this number is comparable from Bower's kungulus ({10,100,3} & 10).
So grand penthrathoth is equal to E100#^^^#^#100#2 (This number is comparable from Bower's kungulusplex ({10,kungulus,3} & 10)), grand grand penthrathoth is equal to E100#^^^#^#100#3 and granpenthrathoth is equal to E100#^^^#^#100#100,
Ok, here more.
greapenthrathoth = E100#^^^#^#100#100#100
giganpenthrathoth = E100#^^^#^#100#100#100#100
gorgepenthrathoth = E100#^^^#^#100##5
...
gugolpenthrathoth = E100#^^^#^#100##100
graatagolpenthrathoth = E100#^^^#^#100##100#100
greegolpenthrathoth = E100#^^^#^#100##100#100#100
...
gugoltriapenthoth = E100#^^^#^#100##100##100
gugoltelsapenthoth = E100#^^^#^#100##100##100##100
gugolpentapenthoth = E100#^^^#^#100##100##100##100##100
throopenthrathoth = E100#^^^#^#100###100
teroopenthrathoth = E100#^^^#^#100####100
godgahpenthrathoth = E100#^^^#^#100#^#100
tethrapenthrathoth = E100#^^^#^#100#^^#^#100
penthratrithoth = E100#^^^#^#100#^^^#^#100
deutero-penthrathoth = E100#^^^#^#*#^^^#^#100
trito-penthrathoth = E100#^^^#^#*#^^^#^#*#^^^#^#100
penthrathothatria = E100#^^^#^^^#^#100.
penthrathothatetra = E100#^^^#^^^#^^^#^#100
penthrathothapenta = E100#^^^#^^^#^^^#^^^#^#100
Hexathrathoth Group
A hexathrathoth = E100#^^^^#^#100. This number is comparable from Bower's quadrunculus ({10,100,4} & 10).
hexathrathothatria = E100#^^^^#^^^^#^#100. This number is comparable from my quadrunculusplex.
Quadrunculusplex = {10,quadrunculus,4} & 10 and has geegolplex entries.
hexathrathothatetra = E100#^^^^#^^^^#^^^^#^#100
A hepthrathoth = E100#^^^^^#^#100. This number is comparable from my quintunculus.
Quintunculus = {10,100,5} & 10 and has gigol entries.
hepthrathothatria = E100#^^^^^#^^^^^#^#100
hepthrathothatetra = E100#^^^^^#^^^^^#^^^^^#^#100
A octhrathoth = E100#^^^^^^#^#100. This number is comparable from my sextunculus.
Sextunculus = {10,100,6} & 10 and has goggol entries.
octhrathothatria = E100#^^^^^^#^^^^^^#^#100
octhrathothatetra = E100#^^^^^^#^^^^^^#^^^^^^#^#100
A nonthrathoth = E100#^^^^^^^#^#100. This number is comparable from my septunculus.
Septunculus = {10,100,7} & 10 and has gagol entries.
nonthrathothatria = E100#^^^^^^^#^^^^^^^#^#100
nonthrathothatetra = E100#^^^^^^^#^^^^^^^#^^^^^^^#^#100
...
Keep a up!
Omegathrathoth Group
Now beyond!
Godgathor = E100#^#^#100
Tethrathoth = E100#^^#^#100
Penthrathoth = E100#^^^#^#100
Hexathrathoth = E100#^^^^#^#100
Hepthrathoth = E100#^^^^^#^#100
Octhrathoth = E100#^^^^^^^#^#100
Nonthrathoth = E100#^^^^^^^^#^#100
Decthrathoth = E100#^^^^^^^^^#^#100
Undecthrathoth = E100#^^^^^^^^^^#^#100
...
Omegathrathoth = E100#^^...^^#^#100 \w 100 #'s. This number is comparable from Bower's humongulus ({10,10,100} & 10).
Wait, there a bigger number, Omegathrathothigong = E100000#^^...^^#^#100000 \w 100000 #'s
Wait, there MORE bigger, Grand Omegathrathoth = E100#^^...^^#^#100 \w omegathrathoth #'s. This number is comparable from my humongulusplex.
Humongulusplex = {10,10,humongulus} & 10 and has {10,10,humongulus} entries.
Wait, there and a biggest Nested Cascading-E notation number, Grand Omegathrathothigong = E100000#^^...^^#^#100000 \w Omegathrathothigong #'s.
What? This is limited of NCEN! We continue this, using xE^...?!
Granomegathrathoth = E100#$#^#100#100
Gugolomegathrathoth = E100#$#^#100##100
Throomegathrathoth = E100#$#^#100###100
Godgahomegathrathoth = E100#$#^#100#^#100
Tethraomegathrathoth = E100#$#^#100#^^#^#100
Penthraomegathrathoth = E100#$#^#100#^^^#^#100
Omegathraomegathrathoth = E100#$#^#100#$#^#100
Omegathraomegathraomegathrathoth = E100#$#^#100#$#^#100#$#^#100
Until ...-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathra-omegathrathoth.
Let called Extended Nested Cascading-E Notation. This is still not far enough.
As NEW Expand-E Notation (E^<#>), ^ change to <#>, ^^...^^ (N ^'s) change to <##...##> (N #'s), and ^^...^^ (X ^'s) or $, using Extended Nested Cascading-E Notation change to <#^#>.
As Extensible-E System, Expand-E Notation are fifth term.
(Modified Version)
***Later Aarex modified the article and deleted most of his googolisms.
As a Cascading-E Notation, he reaches a Great and Terrible Tethrathoth.
So here a extension of Cascading-E Notation:
A new rules is Uses a @^...^# = #^...(#^...^(#...)).
Nested Cascading-E Notation (E^...)
Other name "Cascading Hyper-E Notation"
E(a)b = a^b
# 1 = #
# n#m = # (# n#m-1)
# #1m = # #m
# n #l m = # n #l-1 n...n #l-1 n (m n's)
# n @^# m = # n @*@...@*@ m (n @'s)
# n @^^...^^# m = n @^^...^@...@^^...^@ n (m-1 ^^...^'s)
# n @^...^(@ #) m = # n (@^...^@)^^# m
#^# --> X
#^#*# --> X+1
#^#*## --> X+2
#^#*### --> X+3
...
#^#*#^# --> 2X
#^#*#^#*# --> 2X+1
#^#*#^#*## --> 2X+2
...
#^#*#^#*#^# --> 3X
...
#^#*#^#*#^#* ... *#^#*#^# w/n #^#s --> nX
Next we have #^## --> XX = X2. So the #^## has the same power as planar arrays. Continuing on we have:
#^## --> X2
#^##*# --> X2+1
#^##*## --> X2+2
...
#^##*#^# --> X2+X
#^##*#^#*# --> X2+X+1
...
#^##*#^#*#^# --> X2+2X
#^##*#^#*#^#*#^# --> X2+3X
...
#^##*#^## --> 2X2
#^##*#^##*#^## --> 3X2+X
...
#^### --> X3
#^#### --> X4
...
#^^## = #^#^# --> XX
Here we observe an interesting property of the comparison. The number of hashes in the exponent is also the number of dimensions of the comparable array. We can say if we have #^$(#) where $(#) is some expression composed of hyperions, replace all of these with X and we can say:
#^$(#) --> $(X)
Hence...
#^^### = #^#^#^# --> X^X^X
#^^#### = #^#^#^#^# --> X^X^X^X
#^^##### = #^#^#^#^#^# --> X^X^X^X^X
...
This notation reaches a #^^#, so here more:
#^^#^# --> X^^X
(#^^#)^# --> (X^^X)^X
(#^^#)^(#^^#) --> (X^^X)^(X^^X)
(#^^#)^(#^^#)^(#^^#) --> (X^^X)^(X^^X)^(X^^X)
#^^## --> ~ X^^2X
#^^### --> ~ X^^3X
#^^#^# --> ~ X^^(X^2)
#^^#^## --> ~ X^^(X^3)
#^^#^#^# --> ~ X^^(X^X)
#^^#^#^#^# --> ~ X^^(X^X^X)
#^^#^^# --> ~ X^^X^^X
#^^#^^#^^# --> ~ X^^X^^X^^X
How about pentation?
#^^^# --> ~ X^^^X
#^^^## --> ~ X^^^2X
#^^^#^# --> ~ X^^^X^2
#^^^#^^# --> ~ X^^^X^^X
#^^^#^^^# --> ~ X^^^X^^^X
And hexation, heptation, ...?
#^^^^# --> ~ X^^^^X
#^^^^^# --> ~ X^^^^^X
In FGH, this growth rate is fφ(ω,0)(n) > E(n)n#^...^#n /w n+1 ^'s.
As Extensible-E System, Nested Cascading-E Notation are fourth term.
As Hyper-E Notation is first (E#), Extended Hyper-E Notation (xE#) is second and Cascading-E Notation (E^) are third.
Formly New E#^...
Get ready to reaching #^...^#...
Tethrathoth Group (Expanded)
Now revisit of tethrathoth:
Tethrathoth = E100#^^#100
Did you know: #^^# in Extended Cascading-E Notation is same!
Grand Tethrathoth = E100#^^#100#2
Grantethrathoth = E100#^^#100#100
Greatethrathoth = E100#^^#100#100#100
Gigantethrathoth = E100#^^#100##4
Gorgetethrathoth = E100#^^#100##5
Gultethrathoth = E100#^^#100##6
Gasptethrathoth = E100#^^#100##7
Ginortethrathoth = E100#^^#100##8
There a E100#^^#100##100 exist, so
gugold ~ tethrathoth =
Gugoltethrathoth = E100#^^#100##100
throogol ~ tethrathoth =
Throotethrathoth = E100#^^#100###100
Tristo-Throotethrathoth = E100#^^#100###100
teroogol ~ tethrathoth =
Terootethrathoth = E100#^^#100####100
Petootethrathoth = E100#^^#100#####100
Ectootethrathoth = E100#^^#100######100
We defined tetrithrathoth = E100#^^#100#^^#100
teterthrathoth = E100#^^#100#^^#100#^^#100
Deutero-tethrathoth = E100#^^#*#^^#100
Trito-tethrathoth = E100#^^#*#^^#*#^^#100
This is end of expanded tethrathoth group.
Terriblethrathoth Group
To define Terriblethrathoth, we need #^^## seperator. E100#^^##100 = E100(#^^#)^^#100 = E100(#^^#)^(#^^#)...(#^^#)^(#^^#)100 (with 100 #^^#'s) = ...
So Terriblethrathoth (old is Terrible Tethrathoth) is equal to E100#^^##100.
Then Thrthrathoth (old is Terrible Terrible Tethrathoth) = E100#^^###100
Terthrathoth (old is Terrible Terrible Tethrathoth) = E100#^^####100
Pethrathoth (old is Terrible Terrible Terrible Tethrathoth) = E100#^^#####100
Ecthrathoth = E100#^^######100
Zetthrathoth = E100#^^#^#7
And Yotthrathoth = E100#^^#^#8.
Tethrathoth ba'al is E100#^^#^#100 and Great and Terrible Tethrathoth is E100#^^#^#100#2.
Wait, there more!
Great and Terrible Tethrathoth ba'al is E100#^^#^#100#2
Great and Terrible Tethrathoth super ba'al is E100#^^#^#100#3
Great and Terrible Tethrathoth duper ba'al is E100#^^#^#100#4
Great and Terrible Tethrathoth truper ba'al is E100#^^#^#100#5
...
Tethrathor Group
Tethrathor = E100#^^#^^#100
Terriblethrathor = E100#^^#^^##100
Thrthrathor = E100#^^#^^###100
Review
We need new numbers!
Let called Extended Nested Cascading-E Notation. This is still not far enough.
As NEW Expand-E Notation (E^<#>), ^ change to <#>, ^^...^^ (N ^'s) change to <##...##> (N #'s), and ^^...^^ (X ^'s) or $, using Extended Nested Cascading-E Notation change to <#^#>.
As Extensible-E System, Expand-E Notation are fifth term.
Okay, as extension of Nested Cascading-E Notation, here it a Expand-E Notaion!
Expand-E Notation (E^<#>)
Here a new rules:
@ a #^...<@ #>@ b = @ a #^...^<@>@ b
#^...^<>@ = #^...^@
@ a #^...^@ b = @ a #*@ b, if 0 ^'s
#<#^#>#^# = X^...^X (X ^'s)
#<#^#>#<#^#>#^# = X^...^X^...^X (X ^'s between X)
#<#^#>#<#^#>#<#^#>#^# = X^...^X^...^X^...^X (X ^'s between X)
#<#^#*#>#^# = X^...^X (X+1 ^'s)
#<#^#*#^#>#^# = X^...^X (X*2 ^'s)
#<#^#*#^#*#^#>#^# = X^...^X (X*3 ^'s)
Omegathrathoth = E100#<#^#>#^#100
Grand Omegathrathoth = E100#<#^#>#^#100#2
Granomegathrathoth = E100#<#^#>#^#100#100
Greaomegathrathoth = E100#<#^#>#^#100#100#100
Gugolomegathrathoth = E100#<#^#>#^#100##100
Throomegathrathoth = E100#<#^#>#^#100###100
Godgahomegathrathoth = E100#<#^#>#^#100#^#100
This is extension of Expand-E Notation. Array-E Notation (E#<...>)
I added 2 rules:
@ a #<,...,#...## @>#...## b = @ a #<,... #<,...,#...## @>#...#,#...# @>#^# b
@ a #<...,#>#...# b = @ a #<...>#...# b