I know of various properties of numbers that are known to have no largest element (like natural numbers, primes, etc.) and I know of unproven conjectures that certain properties have a largest element (twin primes, etc.) but I don't remember examples of properties that have been proved to have a largest element, at least within number theory. I suspect there are some, so I thought I'd ask here.
Outside of number theory, I know of some geometric ones -- like the number of sides in a Platonic solid.
What's Special About This Number? may be a useful source.