TITLE: Properties of natural numbers such that there is a "very large largest" number with that property
QUESTION [12 upvotes]: 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.

REPLY [5 votes]: A good start might be finite sequences in OEIS. The search keyword:  fini returns 
4660 results including Gerry's narcissistic number and left-truncatable prime.
Another example is A080601 Number of positions that the 3 X 3 X 3 Rubik cube puzzle can be in after exactly n moves with largest term $91365146187124313$. 
The search can be automated filtering only large numbers.
Probably you are not interested in this, but artificial solutions can be constructed easily:
Solutions $x$ of $x^2-y^2=\text{big}$ or trivially roots of $p=\prod_{i \in S} (x-i)$.