I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a topological space $X$, which leads one to the notion of stacks etc. The answer explained why a certain "sheaf of groupoids" naturally arises when considering the presheaf of isomorphism classes of principal $G$-bundles on a space.

I'm not that far into my study of algebraic geometry/topology but recently I have been working quite hard trying to understand sheaves at various levels of generality as I find them really interesting in themselves. So are there other contexts in which we might require something like a sheaf of categories/groupoids on a space?