Here is an example appearing 'in nature': the category $\mathbf{Sch}$ of schemes has arbitrary (small) coproducts (disjoint unions), but existence of (small) products is a subtle question (on the other hand, finite limits do exist, whereas finite colimits are subtle). For instance, the product $(\mathbf P^1_{\mathbf Z})^{\mathbf N}$ does not exist in $\mathbf{Sch}$; see [Stacks, Tag 078E].
As noted before, to get an example where products exist but coproducts do not, take the opposite category $\mathbf{Sch}^{\text{op}}$.