Exceptions are interesting. Sometimes, they're also important. If a theorem with exceptions is important for a subject, there are liable to be many corollaries of the form "either this is true... or these exceptional circumstances hold". When the exceptions are finite in number, it's particularly nice because then it's feasible to build a complete picture of the exceptions.

Many exceptions turn out to be related to one another, or to other important ideas. There are plenty of examples on Wikipedia, which has an impressive diagram relating some of them.

Here are a few more, in no particular order:

• The hyperbolic plane can be tiled with $n$-gons, three meeting at a vertex, iff $n > 2\pi$.

• Every prime is of the form $4n \pm 1$, except $2$.

• The quadratic field $\mathbb{Q}[\sqrt{-n}]$ has unique factorization in its ring of integers iff $n \in \{1,2,3,7,11,19,43,67,163\}$, the Heegner numbers.

• The finite ring $\mathbb{Z}/n\mathbb{Z}$ admits a compatible exponential operation iff $n \in \{1,2,6,42,1806\}$, with the sequence ending there because $1807$ is composite.

• Surgery theory works in dimensions $>4$ (for smooth manifolds) or $>3$ (for topological manifolds).

I'd also welcome examples where the set of exceptions is not literally finite but is relatively compact, given a sensible (non-compact) topology on the overall space. For example:

• If $0 \le x < y$ then either $x^y > y^x$ or $x < \mathrm{e}$.