is there a (useful) finite number that is so big we haven't been able to formulate an upper bound for it?
@SuricrasiaOnline Busy beaver numbers come to mind. There is no computable function f such that f(n) > BB(n) for all n.
@SuricrasiaOnline When you start invoking Ackerman functions or chained-arrow notation I tend to assume we've left the realm of "(useful)", so I'm going to go with "no", but I'm sure there are grad students who would disagree.
It depends how good you are in math.
In my case that's close to 241.
Once i counted to 238 and i'm pretty sure this algorithm could continue at least to 241 but not much.