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Great explanation, but I'm somewhat puzzled by the fact that you never actually told us how Graham's number is defined. It's like there's a missing paragraph after you finish explaining the rapid recursive growth of up-arrows.

As I understand it, you start with 3^^^^3,

i.e. 3^^^(3^^^3)

i.e. 3^^(3^^(3^^3))

i.e. 3^^(3^3^3^3^3^3^3...) where "..." continues on for 3^3^3 iterations of powers of three

i.e. 3 raised to the third power a number of times equal to (3^3^3^3^3^3^3^3...) from above

i.e. already an unimaginably large number.

You start with the end result of that, then put that many *up arrows* in between two threes. That is to say that after we just got that huge number result that I can't concisely describe from just 3^^^^3, we look at the number 3^^^^^...3 that has that indescribable number of ^'s in it. That's "Step 1".

Then you do it again; take the number from Step 1, and put *that many* ^'s between two threes. Well beyond numbers anyone can really hope to imagine in any meaningfully representative way without deep mathematical understanding, this completes Step 2.

Graham's Number is the result when we reach Step 64, each of these steps involving putting a number of ^'s equal to the value of the previous step in between two threes.

Anyway, that's what I think an explanation might look like; I'm surprised something like this wasn't included. Otherwise, great explanation,