This integer only exists if you assume classical logic. Otherwise, there is no such integer a priori, and actually there is none in general.
For that to be the case, there would have to be some symbol under the tape and some state the machine is in for which the action the machine takes and the next state it goes to would depend on the axioms taken somehow. There is no place where the Turing Machine has somehow been running for so long and just gotten so large that its behavior becomes non-deterministic somehow.
What this means is that even if we lived in a universe where we had the unfathomable resources to actually have this number somehow meaningfully "in hand", we would be unable to prove that it was the correct one with just ZFC. Maybe one of the really quite numerous other machines still spinning away would in fact terminate in the future and be the real BB winner, because even this staggeringly monstrously large universe is still piddling nothing next to infinity and the other machines still require infinite resources to run them to discover they never terminate. But that doesn't do anything to affect whether or not there in fact is a single concrete integer that corresponds to BB(748).
Although one imagines that any universe with the resources to "have" BB(748) in it might also have some much more powerful axiom systems to play with in the process. The amount of computational power this universe apparently possesses is beyond all comprehension and who knows what they could know. But even if they used a more powerful system, it wouldn't change what BB(748) is... it just might affect whether or not they were correct about it.
That's easy, you just have to be an ultrafinitist, and say, "The definition of a TM presupposes an infinite set of natural numbers for time steps and tape configurations. But there aren't actually infinitely many natural numbers, infinitely long executions, arbitrarily long proofs, etc., outside of the formalism. If a formal statement and its negation do not differ regarding any natural numbers small enough to actually exist (in whatever sense), then neither is more true than the other." In particular, consistency statements may have no definite truth value, if the hypothetical proof of an inconsistency would be too large.
Of course, metamathematics tells us "you can't do that, in principle you could tell the lie if you wrote out the whole proof!" But that principle also presupposes the existence of arbitrarily-long proofs.
(Personally, hearing some of the arguments people make about BB numbers, I've become attracted to agnosticism toward ultrafinitist ideas.)
Plus, ultafinitism is essentially relative to the universe you find yourself in. I hypothesized a universe in which BB(748) could actually exist, but you can equally hypothesize ones in which not only can it exist, it exists comfortably and is considered a small number by its denizens. We can't conceive of such a thing but there's no particular a priori reason to suppose it couldn't exist. If such a universe does actually "exist" does that mean our ultrafinitism is wrong? I'm actually a sort of a proponent of knowing whether your operating in a math space that corresponds to the universe (see also constructive mathematics), but concretely declaring that nothing could possibly exist that doesn't fit into our universe is a philosophical statement, not a mathematical one.
The formalism says that there's still a definite integer that it corresponds to. The ultrafinitist would deny that the formalism keeps capturing truth past where we've verified it to be true, or some unknown distance farther.
> I hypothesized a universe in which BB(748) could actually exist, but you can equally hypothesize ones in which not only can it exist, it exists comfortably and is considered a small number by its denizens.
Sure, but the ultrafinitist would argue, "All this is still just a shallow hypothesis: you've said the words, but that's not enough to breathe much 'life' into the concept. It is but the simplest of approximations that can fit into our heads, and such large things (if they could exist) would likely have an entirely different nature that is incomprehensible to us."
> We can't conceive of such a thing but there's no particular a priori reason to suppose it couldn't exist.
That's why I wouldn't call myself an ultrafinitist, but would prefer an agnostic approach. There may be no great a priori reason to suppose it cannot exist, but I similarly do not see any such reason it must necessarily exist. We empirically notice that our formalism works for numbers small enough to work with, and we pragmatically round it off to "this formalism is true", but one could argue that surprising claims about huge numbers need stronger support than mere pragmatism.