Aaronson is talking about Graham‘s number when he writes: “Recondite as it seems, the Ackermann sequence does have some applications. A problem in an area called Ramsey theory asks for the minimum dimension of a hypercube satisfying a certain property. The true dimension is thought to be 6, but the lowest dimension anyone’s been able is prove is so huge that it can only be expressed using the same ‘weird arithmetic’ that underlies the Ackermann sequence. Indeed, the Guinness Book of World Records once listed this dimension as the biggest number ever used in a mathematical proof.”

That‘s about a third of the way through the article. The numbers get a lot bigger after that.

As a footnote, the true dimension is no longer thought to be six — Aaronson’s essay was written in 1999 — and is now known to be at least 11. The Wikipedia article linked above has a reference for this improved lower bound.

Let g = Graham's number. Then use Knuth's up arrow notation. Then (and this is perhaps cheating) use a non-computable function like the busy beaver function, which grows faster than ANY computable function. That last step may in fact not be cheating, because it might be possible to prove a lower bound and therefore prove that it's higher than the opponent's number.

Pretty notation here:

http://mathbin.net/50035