What are some properties proved to have largest elements?

Question

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.

Answer 1

Here are two examples:

Factorions are numbers that are the sum of the factorial of their digits. For instance, $145 = 1!+4!+5!= 1+24+120=145$. The largest factorion is 40585.

Numbers that are the cube of their digit sum, such as $512 = (5+1+2)^{3} = 8^{3}$. The largest is 19683.

Answer 2

The set $\{n\in\mathbb{N}\,|\,n\text{ and }n+1\text{ are perfect powers}\}$ has a largest element: $8$. Actually, it's its only element.