No. The busy beaver function grows faster than any computable function.

What's more, for any N > 549, you cannot prove (in the usual universe of mathematics) that any computable number is equal to or larger than BB(N). (At least that's my understanding of the independence results (https://wiki.bbchallenge.org/wiki/Independence_from_ZFC))

The algorithm for BB(N) is (with some hand waving) “for each N-state Turing machine, check if it halts on an initially empty tape. If it does not, skip it. If it does, run it until it halts and count the number of steps, k. BB(N) is the maximum value of k you find.”

The problem is that the naive algorithm above requires an oracle for the Halting Problem, which every CS student learns is impossible in general for a computational model that isn’t more powerful than a Turing Machine.

So if a function can be expressed in terms of a Turing machine, busy beaver has to grow faster because eventually for some value of N, you can just encode the candidate function as a Turing machine and have Busy Beaver simulate it.

Busy Beaver hunters on the low end (trying to find the next number) have to exhaustively prove whether each Turing machine eventually halts or not at that number of states so that they can evaluate the longest running one with finite execution.

TREE(3) is so crazy to me. You would expect it to either be a reasonable number, or to be infinite (i.e. there's an infinite sequence of such trees).

But no. It's just an _impossibly unfathomably large number_. Eventually, there's no more trees with less than $i$ vertices, but only at such a stupid incredibly large $i$ that it makes other "googology" numbers vanish by comparison.

> it makes other "googology" numbers vanish by comparison.

I consider TREE to be medium sized in the googology universe. It obliterates Graham's Number of course, as well as Goodstein sequence and simple hydras. But it's easily bested by an ingenious matrix generalization of hydras known as the Bashicu Matrix System, even when fixed at 2 rows. 3-row BMS goes far beyond that, let alone n-row. BMS is so simple that it can be encoded in under 50 bytes [1], where TREE would take significantly more. Beyond that there vastly greater numbers still such as Loader's Number. I shouldn't mention Rayo's because that's no longer computable in the usual sense.

[1] https://codegolf.stackexchange.com/questions/18028/largest-n...

Busy Beaver is non computable and grows much faster than the various subcubic graph numbers (you could easily encode a subcubic graph number computation in a relatively small turing machine).

The busy beaver function is interesting precisely because you _can't_ come up with any computable function that grows faster.