From Borceaux' Categorical Algebra Vol 1:
Proposition $2.9.4$ Consider a category $\mathcal{C}$ and an object $C \in \mathcal{C}$. The representable functor $\mathcal{C}(C,-) : \mathcal{C} \to \mathsf{Set}$ preserves all existing limits, including large ones. In particular, it preserves monomorphisms.
Which makes me wonder, are there interesting large limits? Large here means the domain category is large. A necessary condition for being interesting is that the large category is not equivalent to a locally small one.
Here's an interesting large colimit:
The "Local State Classifier" was recently introduced in work of Ryuha Hora to solve an open problem of Lawvere in the special case of boolean topoi. In particular, it gives an internal handle on understanding the quotient topoi of a boolean topos in a way analogous to Lawvere-Tierney topologies internally classifying subtopoi. You can read more in Hora's paper Internal Parametrization of Hyperconnected Quotients.
Let $\mathcal{E}_\text{mono}$ be the subcategory of $\mathcal{E}$ were we keep all the objects, but only the monomorphisms. Then if $\iota : \mathcal{E}_\text{mono} \to \mathcal{E}$, Hora defines the local state classifier to be the (large!) colimit of this inclusion functor. There is another characterization which shows that this colimit actually exists (there's no reason for a grothendieck topos to have colimits of this size), but the colimit structure seems (to me) to be quite important for both the intuition and the proof of the main theorem.
I hope this helps ^_^