Here are a slightly different flavor of examples, of course as mentioned by Peter Lumsdain, the trick is always the same looking at categories where $Hom(I,X)$ looks quite different from the "underlying sets of $X$"


Consider the category whose objects are set, and whose morphisms are relations, i.e. subsets of $X \times Y$ (with the usual composition of relation).

It has a monoidal structure given by the product of sets. The exponential for this monoidal structure is simply product, indeed,

A relation from $X$ to $Y \times Z$ is the same as a relation from $X \times Z$ to $Y$.

Other similar examples includes:

- The 2-category of sets and span of sets (with the products of sets as monoidal structure).

- The category of small categories and profunctor, with the products of categories as monoidal structure, here the exponential $[X,Y]$ is given by $Y \times X^{op}$

- The category of polynomial functor (which is actually cartesian closed) described in [The cartesian closed bicategory of generalized species and structures][1] by Fiore,Gambino,Hyland and Winskel. Where the exponential is a little more complicated, but still has the same flavor.




  [1]: https://www.cs.le.ac.uk/people/ngambino/Publications/generalised-species.pdf