Nice examples of finite things which are not obviously finite

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:

1. The finiteness of $\text{Gal}(\overline{\mathbf R}/\mathbf R)$;
2. The finite-generatedness of homotopy groups of spheres;
3. The finiteness of the set of smooth structures on the $n$-sphere, for $n\neq 4$;
4. The conjectured finiteness of Shavarevich-Tate groups;
5. The finite-generatedness of Mordell-Weil groups;
6. The finiteness of class numbers;
7. 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?

• Sorry for this very late comment, but I would like to know what $\overline{ \Bbb R}$ is. I don't think this denotes an algebraic closure of $\Bbb R$, because then it would be isomorphic to $\Bbb C$, and the finiteness of $\text{Gal}(\Bbb C / \Bbb R)$ is not hard to establish, in my opinion. – Watson Jun 13 '16 at 11:31
• Moreover (and this is completely unrelated), I found in that question a link to one of your blog post. I was wondering : are some articles of this blog still available anywhere ? – Watson Jun 13 '16 at 11:32

The finiteness of the number of different $n$ for which Fermat's equation has a solution?