Greagol

The Greagol (short for 'great googol') is equal to E100#100#100, using Hyper-E Notation. The greagol can be described as the 100th member of the "grangol series", where the 1st member is a grangol (E100#100), the 2nd is the grangoldex (E100#grangol), the 3rd is the grangoldudex (E100#grangoldex), ... and in general the nth member of the series, call it S(n), is equal to E100#S(n-1). It follows that:

Greagol = E100#(E100#(E100#( ... ... (E100#(E100#100)) ... ... ))) w/100 "E"s in the expression

A Greagol can also be defined as follows:

Greagol = 1010 10 ... ... 10 10 100     w/ 1010 10 ... ... 10 10 100      w/ ... ... ... ... w/ 1010 10 ... ... 10 10 100      w/100 10s 10s ... 10s

where there are 100 power towers, and the number of tens in each is equal to the power tower to the immediate right. The rightmost power tower contains exactly 100 10s and is equal to the grangol.

Comparison with Gaggol
The Greagol is greater than and comparable to the Gaggol. A Gaggol = E1#1#100. Therefore:

Gaggol = E1#1#100 < E100#100#100 = Greagol

Although it would seem that a Greagol is only slightly larger than a Gaggol, it can be shown that:

GaggolGaggol < Greagol

Where the left sided superscript is shorthand for Gaggol tetrated to the Gaggol. Proving this is a little more tricky that proving that Giggol^Giggol < Grangol, but can still be done. To prove this we observe that:

Gaggol = 1010 10 ... ... 10 10    w/10^^^99 10s

It follows that:

GaggolGaggol = GaggolGaggol Gaggol ... ... Gaggol Gaggol    w/Gaggol Gaggols < Gaggol 2 10

This follows from the fact that (10^^A)^^B < 10^^(AB) and in fact (10^^A)^^B is approximately 10^^(A+B)

Gaggol 2 10 < 1+10^^^9910 10

This follows from the fact Gaggol^2 = 102logGaggol << 10Gaggol = 1010^^^100 = 10( 10^^^9910) = 1+10^^^9910

1+10^^^9910 10 < 1+10^^^9810 10 10 < ... < 1+10^^^110 ... 10 w/100 10s = 1110 ... 10 w/100 10s = E1#11#100 <

E100#100#100 = Greagol.

Furthermore we can provide the following upper and lower bounds for GaggolGaggol: E1#10#100 < GaggolGaggol < E1#11#100