Are there important locally cartesian closed categories that actually are not cartesian closed?

The example most familiar to me is the category $\mathcal{LH}$ whose objects are topological spaces and whose morphisms are local homeomorphisms.

This category doesn't have a terminal object, but it is locally cartesian closed: each slice category $\mathcal{LH}/X$ is equivalent to the category of sheaves on $X$.

Another example of an LCCC without a terminal object but each of whose slices is a (presheaf) topos is the category of finitary polynomial endofunctors in Set. This is very nicely explained in [Kock, 2011].

This is due to the lack of a "universal polynomial endofunctor" and in fact the $\infty$-category of polynomial endofunctors in $\infty$-Gpd is an $\infty$-topos.