What is a non-logical concrete example of a choiceless elementary topos? What is a non-logical concrete example of a choiceless elementary topos?
The elementary topoi made from choiceless models of ZF, or stuff like the realizability topos, are (I believe) choiceless, but is there a concrete example from other branches of mathematics?

Assuming you mean choice in the "every epimorphism has a section" sense, counterexamples among categories of presheaves are plentiful.
For a very quick and silly example, there's the category $\mathbf{Set}^G$ for a non-trivial group $G$. In this case, the sole representable functor in the category is the transitive action of the group on itself, so it has no fixed points; so there is no section of the unique map from this object to the terminal object of $\mathbf{Set}^G$.
Another, similar example is the category $\mathbf{Set}^{\bullet\rightrightarrows\bullet}$ of directed multigraphs. Consider the graph with two vertices, and one arrow in each direction between them; then the unique map from this to the terminal object also doesn't have a section.
More generally, toposes satisfying choice have a classical internal logic, so choosing any topos with a strictly intuitionistic internal logic will involve some failure of choice. You can hardly throw a stone without hitting an example of a topos where choice fails.
