The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a topos. Is that the only way there is?
Well, I suppose there's another way: the category of topological spaces and local homeomorphisms is locally cartesian closed. So one can get a similar effect by throwing out morphisms. But in this example the category is not even accessible.
Let's stipulate that the category should also not be a poset.
After all, a topos is just a locally cartesian closed category with a subobject classifier. Is the subobject classifier merely a representability condition, or does it have additional exactness implications beyond local cartesian closure?