TITLE: Nice examples of finite things which are not obviously finite
QUESTION [6 upvotes]: This question is in the spirit of the question "Nice examples of groups which are not obviously groups".
There are many impressive finiteness results in mathematics. For example:

The finiteness of $\text{Gal}(\overline{\mathbf R}/\mathbf R)$;
The finite-generatedness of homotopy groups of spheres;
The finiteness of the set of smooth structures on the $n$-sphere, for $n\neq 4$;
The conjectured finiteness of Shavarevich-Tate groups;
The finite-generatedness of Mordell-Weil groups;
The finiteness of class numbers;
The finiteness of the set of rational points on a curve $X/\mathbf Q$ when the genus $X>1$...

So, what are the nicest examples of finite sets which are not obviously finite?

REPLY [2 votes]: The finiteness of the number of different $n$ for which Fermat's equation has a solution?