I'm looking for properties (P) such that you **would assume** that there are infinitely many natural numbers with property (P) (for example because there are very large numbers with that property) but where it turns out that there are only finitely many.

EDIT: Note that the eventual counterexamples question asks for (P) such that the **smallest** $n$ with property (P) is large; the current question asks for (P) such that the **largest** $n$ with property (P) is large.