In some (but not all) of the published definitions, a locally cartesian closed category is any category with all its slices cartesian closed. Such a category need not be cartesian closed itself, simply because it need not have a terminal object.
As a prominent example, there is the $n$-category Cafe: https://ncatlab.org/nlab/show/locally+cartesian+closed+category
I see how someone could find this a formally natural definition. In particular, someone might want to de-emphasize terminal objects for purposes of dependent type theory because terminal types are not very useful in programming even though they normally exist.
But I am curious to know if there are important examples of locally cartesian closed categories in this sense that actually do not have a terminal object (i.e. are actually not cartesian closed)?
One source of examples occurs to me: The disjoint union $C+D$ of any two cartesian closed categories $C,D$ is locally cartesian closed.
That does not look important to me, though I do not know it is not. There is a trivial embedding of that disjoint union as a pair of slice categories of a cartesian closed category. Formally adjoin a new terminal object for the whole disjoint union category, and add maps making the new terminal object into the disjoint union of the two previous terminal objects. But I cannot see this as an important kind of example, because to me this just looks like a presentation of the coproduct in the 2-category of toposes: which is realized by the product of the toposes in the 1-category of categories. (The objects $a,b$ of $C$ and $D$ appear as pairs $(a,1)$ and $(1,b)$ in the product, with the original terminal objects $1$.)
Best Answer
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 $X$.