The category $\mathbf{Met}$ of metric spaces and metric maps is another example. There are finite limits, but no infinite products. Countable products exist at least when we use continuous maps as the morphisms instead. $\mathbf{Met}$ has no binary coproducts, and this can be seen as the starting point of the Gromov-Hausdorff distance, where we consider all possible metrics on a disjoint union. Coequalizers do not exist either. It is worth pointing out that the injective objects of $\mathbf{Met}$ have gathered some interest.