The terminal category is a Grothendieck topos which is essentially small. Are there other examples of Grothendieck toposes which are essentially small? Are there other examples of Grothendieck toposes which are countable?
Similar questions have been discussed here and here - the second thread answers my question with "Grothendieck" replaced by "elementary". But I'm asking specifically about Grothendieck toposes.
An essentially small complete (or cocomplete) category must be a poset (proof: if there are two distinct morphisms $A\to B$, then there must be arbitrarily many distinct morphisms from $A$ into products of copies of $B$). An elementary topos which is a poset must be equivalent to the terminal category (for instance, the existence of subobject classifiers in a poset immediately implies every object has only one subobject). So the only essentially small complete or cocomplete elementary topos (in particular, the only essentially small Grothendieck topos) is (up to equivalence) the terminal category.