User blog comment:Cookiefonster/New article on Pointless Large Number Stuff!/@comment-35851058-20190402170133/@comment-4224897-20190403201218

Good question. I avoided writing much about the exact details of these large number sequences in my article, though if I were to write more articles on this topic I'd definitely write more about those details.

TREE(3) has been proven to dominate all functions provably recursive in the system \(\text{ACA}_0\)+\(\Pi_2^1\)-\(\text{BI}\). I'm not familiar with all these advanced logic systems, but I do know this proof puts a lower bound on TREE(3) in the fast-growing hierarchy, with some crazy ordinal—greater than or equal to vartheta(\Omega^\omega\omega), according to this wiki—that far overpowers any upper bounds for Graham's number in that hierarchy. Graham's number is upper bound by the \(f_{\omega+1}(64)\).

You don't have to know what all this means or how it's known; I don't understand all of it either. Just know that systems of logic are the most common way to prove that sequences like this grow so quickly.