Essentially all naturally occurring large categories are complete and cocomplete. The "only" counterexample (in the sense of, say, categories met by a typical undergraduate) is the category of fields. A "random" large category, if such a thing were to exist, has no reason to have any limits or colimits at all, though.
Essentially all naturally occurring large categories are locally presentable, or certainly at least accessible. The "only" counterexample, in the same sense, is the category of topological spaces. Again, a "random" large category should not be controlled by a small set.