Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how about higher category theory, does it remain enough?
Mike Shulman wrote a nice expository paper on set theoretical foundations for category theory
In Section 6 he explains the difficulties of working with large categories using just ZFC, and he discusses various ways to deal with these size issues. Some of these do not assume the existence of an inaccessible cardinal.
I suggest you have a look at Grothendieck's SGA 4, Exposé I, where a lot of category theory is developed (and applied) on the basis of Bourbaki set theory plus Grothendieck's axioms UA and UB about universes.