Introduction to Extensible-E

The Extensible-E system (ExE) is an array notation invented by Sbiis Saibian, centered around powers of 10. This article will serve as a walkthrough, or an introduction to the notation, meant to give an easy-to-understand explanation for those who don't get it yet. Make sure to read carefully, and really try to interpret what's going on.

Level 1. Hyper-E Notation (E#)
This is the most basic form of Extensible-E, and thus most certainly the place to start off.

The rules are simple:

E[n]a = na

E[n]a#b#c#...#x#y#z#1 = E[n]a#b#c#...#x#y#z (# is called a hyperion)

E[n]a#b#c#...#x#y#z = E[n]a#b#c#...#x#(E[n]a#b#c#...#x#y#z-1)

Here the [n] means "in base n". If it is not there, it defaults to base 10, and Ea becomes 10a. The default is used much, much more often than any modified base, and as the expressions become larger and larger, the difference quickly become much less significant.

Examples
E100 = 10100 = googol (10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000)

En = 10n (n-plex)

E100#2 = E(E100) = 1010 100 = googolplex

E100#100 = E(E(E(...(E(E(E100)))...))) with 100 "E"s = 1010 10 ... 10 10 100    with 100 "10"s = grangol

Ea#b = 1010 10 ... 10 10 a    with b "10"s

E100#100#2 = E100#(E100#100) = E100#grangol = grangoldex

E100#100#100 = E100#(E100#(E100#(...(E100#(E100#(E100#100)))...))) with 100 "100#"s = greagol

E100#100#100#100 = E100#100#(E100#100#(E100#100#...(E100#100#100)...))) with 100 "100#100#"s = gigangol

etc., you should notice a pattern here

bonus: E1#1#1#...#1#1#1#a with b "1"s = 10↑↑↑...↑↑↑a with b+1 arrows

Extended Hyper-E Notation (xE#)
This is where the fun begins.

In this, concatenated hyperions are introduced.

However, to denote an undefined amount of hyperions at once, we will use (#^n). (Saibian himself uses carets instead of superscripts)

The rules regarding multi-hyperions are:

E(...)a(#^c)b = E(...)a(#^c-1)a(#^c-1)a(#^c-1)...(#^c-1)a(#^c-1)a(#^c-1)a with b "a"s

All other rules remain unchanged.

Examples
E100##2 = E100#100 = grangol

E100##100 = E100#100#100#...#100#100#100 with 100 "100"s = gugold

E100##100#2 = E100##gugold = gugolda-suplex

E100##100#100 = E100##(E100##(...(E100##(E100##100))...)) with 100 "100##"s = graatagold

E100##100#100#100 = E100##100#(E100##100#...(E100##100#100)...)) = greegold

etc.

E100##100##100 = E100##100#100#...#100#100#100 with 100 "100"s after the "##" = gugolthra

E100###100 = E100##100##100##...##100##100##100 with 100 "100"s = throogol

E100####100 = E100###100###100###...###100###100###100 with 100 "100"s = tetroogol

etc.

Cascading-E Notation (E^)
The notation starts to become really complex at this point, so hang in there.

Remember #^n? Here, that operator actually has use, except now it's #^#.

Here's the definition...

Ea#^#b = Ea####...####a with b hyperions

("#^#b" always means "(#^b)a")

Ea#^#*#b = Ea#^#a#^#a#^#...#^#a#^#a#^#a with b "a"s (remember the previous rules!!)

Ea#^#*(#^c)b = Ea#^#*(#^c-1)a#^#*(#^c-1)...#^#*(#^c-1)a#^#*(#^c-1)a with b "a"s

Ea#^(#^c)b = Ea#^(#^c-1)*#^(#^c-1)*...*#^(#^c-1)*#^(#^c-1)a with b "#^(#^c-1)"s

Ea#^(#^c)*#b = Ea#^(#^c)a#^(#^c)a#^(#^c)...#^(#^c)a#^(#^c)a#^(#^c)a with b "a"s

All other rules remain unchanged.

Told you it would get complex. If you can't keep up at this point, that's understandable.

Examples
E100#^#2 = E100##100 = gugold

E100#^#100 = E100####...####100 with 100 hyperions = godgahlah

E100#^#100#100 = E100#^#(E100#^#(...(E100#^#(E100#^#100))...)) with 100 "100#^#"s = grandgahlah

E100#^#100##100 = E100#^#100#100#...#100#100#100 with 100 "100"s after the "#^#" = gugoldgahlah

E100#^#*#3 = E100#^#100#^#100 = E100#^#100####...(100 hyperions)...####100 = gotrigahlah

E100#^#*#100 = E100#^#100#^#100#^#...#^#100#^#100#^#100 with 100 "100"s = godgoldgahlah

E100#^#*##100 = E100#^#*#100#^#*#100#^#*#...#^#*#100#^#*#100#^#*#100 with 100 "100"s = godthroogahlah

E100#^##2 = E100#^#*#^#100 = E100#^#*####...(100 hyperions)...####100 = deutero-godgahlah

E100#^##100 = E100#^#*#^#*#...*#^#*#^#100 with 100 "#^#"s = gridgahlah

E100#^###100 = E100#^##*#^##*#...*#^##*#^##100 with 100 "#^##"s = kubikahlah

E100#^#^#100 = E100#^####...(100 hyperions)...####100 = godgathor

etc.

Extended Cascading-E Notation (xE^)
(note from initial editor: it's at this point that I myself have no idea what's going on beyond Ea#^^#b = Ea#^#^#^...^#^#^#a with b hyperions. Can someone else with a full understanding of Extensible-E beyond this point please finish this? And also maybe improve what's been done so far, I feel like I did a sloppy job of explaining everything in general. Thanks. - user ArtismScrub)