Does anyone has a simple example of a 1-category $\mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $\mathcal{C}\left[W^{-1}\right]$ is not a 1-category?

Of course there are obvious “big” examples like CW complexes / derived categories, I’m looking for a small example that I’ll be able to understand combinatorially.

Thanks!