Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large. One of many examples of a large category is Set, for Russell's paradox reasons. A category C is locally small if the class of morphisms between any two of its objects is a set. Of course, a small category is necessarily locally small. The converse is not true, as Set is a counterexample.
Now, I can construct categories that are not locally small. However, what's the most common or most reasonable such category?