For the purpose of the following question, a "subtopos" of a topos $\mathcal{E}$ means a (not necessarily full) replete subcategory of $\mathcal{E}$ that is itself a topos and for which the inclusion functor is a logical functor.
Question: With the above definition of "subtopos", do "non-full subtoposes" exist?
It is easy to come up with full examples, of course. For example, $\mathbf{FinSet}$ is a "subtopos" of $\mathbf{Set}$, and for any group $G$, the $G$-sets with the trivial action form a "subtopos" of the topos of all $G$-sets.
What is not so easy is to come up with non-full examples. For any topos $\mathcal{E}$, there is of course a diagonal functor $\mathcal{E} \to \mathcal{E} \times \mathcal{E}$, which is logical, faithful, and injective on objects, but its image will generally not be a replete subcategory of $\mathcal{E} \times \mathcal{E}$. Also, even if we were to consider the "repletion", which is formed by throwing in all isomorphisms whose domain or codomain is in the subcategory, we still would not get a "non-full subtopos".