I like the example of the non-Grothendieck $\mathbf{Set}$-topos (i.e. there is a non-bounded geometric morphism to $\mathbf{Set}$) given by the category of continuous actions of a proper class-sized topological group. Since this is foundationally a bit subtle, either you take a Grothendieck universe, and work relative to that, or otherwise one can carefully write out a definition or even a characterisation of the underlying category without having to resort to having such data as a topology on a proper class. For instance, one could look at the axiomatisation of what it means for a topos to be the category of continuous actions of a topological group, but then remove the requirement for a generating set.

But other more ad-hoc constructions are sometimes possible. For example, if the topological group is the limit indexed by $\mathrm{ORD}$ of a sequence of open surjections between topological groups, then there's a very nice description in terms of a colimit of the underlying lex cocomplete categories of the continuous actions of these topological groups, and inverse image functors.