Usually, essentially small categories do not have many infinite limits / colimits. For example, the category of finite sets (with all functions) has no infinite coproducts (which, by the way, is a stronger statement than the trivial observation that finite sets aren't closed under coproducts of sets.) But sometimes, somewhat unexpectedly, some infinite limits / colimits do exist.
Example 1. In the category of finite abelian groups, the sequence $\mathbb{Z}/2 \hookrightarrow \mathbb{Z}/2^2 \hookrightarrow \mathbb{Z}/2^3 \hookrightarrow \cdots$ has a colimit. It is zero. This shows that the forgetful functor $\mathbf{FinAb} \to \mathbf{Ab}$ does not preserve colimits, by the way (even though it preserves all coproducts and coequalizers!). The colimit in $\mathbf{Ab}$ is $\mathbb{Z}/2^{\infty} = \mathbb{Z}[1/2] / \mathbb{Z}$.
Example 2. The category $\mathbf{FS}$ of finite sets and surjections has sequential colimits: Every sequence $X_0 \to X_1 \to \cdots$ has a colimit. This is because every such sequence stabilizes: there is some $n_0$ such that all $X_n \to X_{n+1}$, $n \geq n_0$ are isomorphisms. Actually, $\mathbf{FS}$ has all connected colimits!
Let us create a big list of such examples - for educational purposes. In case of essentially small subcategories of (co)complete categories, they shall illustrate in particular that forgetful functors do not necessarily preserve limits or colimits. Example 1 illustrates well that one shall be careful when making claims about the non-existence of limits and colimits. Example 2 is of a different category, since here every diagram stabilizes in some way. More examples of this kind are also appreciated.
