I am curious whether there has ever been a proof which implies the existence of an example or counterexample at some finite value, but one whose size is utterly unknown, and ideally suspected to be very large.
I know there are some proofs that show relatively large numbers exist but we haven't narrowed down where, and even some upper bounds like $e^{e^{e^C}}$. And there's Graham's number, which is the largest number used in a serious proof that I'm aware of, but it's still describable, however abstractly. Since those are all well-defined or at least bounded, they're not what I'm asking for here. Busy Beaver numbers are also not what I'm shooting for.
My original motivation for wondering about this question was in considering the infinitude of polynomial-generated primes; if it turned out that there could only be finitely many primes per any polynomial (does anybody serious consider this a possibility?), it seems to me like the largest primes for most polynomials would have to be laughably or even inaccessibly large, to have somehow completely given no indication yet of that trend whatsoever throughout all the empirical work and heuristics we've amassed.
If anyone can point me towards any proof where an example or counterexample has been shown to exist, yet no bounds are known for a specific value, I'll consider this answered.