I'm willing to dispute your statements that A(G,G) is smaller than your number.
Given that a fairly small set of values for the Ackermann Function can and will give large values, say A(4,1) (which equals 65533, or (2^2^2^2)-3, or 2 tetrated to 3 minus three), A(G,G) can also be written as A(G-1, A(G, G-1)), which can also be written as A(G-2, A(G-1, A(G, G-1)-1)). This holds no real significance, I just found it interesting.
A less confusing way of representing the equation would be 2(↑^(G-2)) (G+3)-3. I'd make that more legible, but I honestly don't know how to do that thing to make the math functions as seen on Wikipedia/Mediawiki.
Because of how Knuth's up-arrow notation works, there would be G-2 up-arrows - since one arrow takes the first number to the power of the second, two tetrates the first to the second, and so on and so fort, that would lead to 2 being taken to the hyper-G of (G-3), with 3 subtracted from that.
That is a very, very large number. It dwarfs Graham's Number by a long shot. If tetration (or hyper-4, its other name) can develop large numbers, with pentation (hyper-5) creating even larger numbers, hyper-G will easily create obnoxiously large numbers, even if the base is infinitesimally larger than 1. Since A(G,G) is sometimes called the "xkcd number", I shall call hyper-G the xkcd hyperoperation.
EDIT: Miscalculated how up-arrows relate to hyperoperators.