Let $C$ be a small category, and consider the class of diagrams $G:D\to C$, with $D$ a small category, that have colimits in $C$. This is a proper class even when $C$ is very small, e.g. whenever $D$ has a terminal object $t$, any functor $G:D\to C$ has a colimit $G(t)$, and there is a proper class of small categories with a terminal object.
However, those colimits feel kind of "trivial"; in some cases at least we can find a small set of diagrams that "carry all the nontrivial information" about colimit diagrams in $C$. For instance, if $C$ is a poset, then it suffices to consider injective functors $G$ (and we may as well take $D$ to be discrete as well), and these form an essentially small set. For a non-posetal $C$ we can't restrict to injective functors, since coproducts are not idempotent, but maybe there is some other restriction that works. Note that by Freyd's theorem, a non-posetal small category does have a bound on the cardinality of coproducts that it can admit; but this doesn't quite answer the question itself, since a particular colimit can exist even if the coproducts that would be necessary to construct it from coproducts and coequalizers do not.
Here are two ways to make the question precise:
Given a small category $C$, is there a small set $L$ of diagrams $G:D\to C$ with colimits such that for any diagram $G':D'\to C$ with a colimit, there is a $(D,G)\in L$ and a final functor $F:D\to D'$ such that $G = G' \circ F$?(Edit: As pointed out by Dylan in the comments, this version is impossible. Take $C$ terminal and let $D$ vary over all ordinals; no small set of categories can be cofinal in all ordinals.)
Given a small category $C$, is there a small set $L$ of diagrams $G:D\to C$ with colimits such that if a functor $H:C\to E$ preserves colimits of all diagrams in $L$, then it preserves all colimits that exist in $C$?
Any solution to question 1 is also a solution to question 2, but I'm not sure whether the converse holds. The mention of Freyd's theorem above suggests that a solution might require classical logic — I would find it more surprising if such a set existed for a non-posetal small complete category, although I don't immediately see an argument that it cannot.
One can of course also ask similar questions for enriched categories, internal categories, $\infty$-categories, and so on. Bonus points go to an answer that applies more generally in such contexts.