Let $\mathcal{C}$ be a category. We say that $\mathcal{C}$ is locally small if $\mathrm{Hom}_{\mathcal{C}}(A,B)$ is a set for all $A$, $B$ in $\mathcal{C}$.
I can't think of any natural examples of non-locally small categories which are 'obviously' not locally small. We can take $\mathcal{C}$ to have one object, and a morphism for every $x \in V$ (say in ZFC), with composition of morphisms given by the union of two sets, but I can't think when this would ever come up 'naturally'
Are there any natural examples of non-locally small categories which are obviously not locally small?