Let $A$ be a commutative ring. Let $\mathcal{E}$ be the category of finite étale $A$-algebras, i.e. those $A$-algebras $B$ such that

1. $B$ is finitely generated and projective as an $A$-module
2. The kernel $I$ of the multiplication map $\mu\colon B\otimes_AB\to B$ satisfies $I^2=I$.

Then $\mathcal{E}^{\text{op}}$ is an elementary boolean topos (but not a Grothendieck topos, as there are no infinite coproducts). The standard way to prove this is by a long detour into algebraic geometry, but it is a nice example to do it more directly. We can take $A$ to be a field or the ring of integers in a number field, and many facts of Galois theory and algebraic number theory have natural interpretations in terms of the corresponding topoi.