I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that categories *can't* describe? 


To be quite specific, I'd be interested any of these:


a.) Objects that can be described by categories that have properties that can't.


b.) Category equivalents of set-theoretic type limits, like how "the set of all sets" causes problems.


c.) Some type of mathematics so pathological it foils, say, associativity. It doesn't need to be a mathematics that's useful in any sense, just one designed specifically to be impossible to describe with categories.



What can't be described by categories?
I don't quite understand your question, but if you're asking whether category theorists should worry about set-theoretic problems the answer seems to be "sometimes". I'm not an expert in this area, but it seems that people tend to avoid universal constructions like limits over large diagrams, and in other cases, people assume the existence of strongly inaccessible cardinals. This seems to avoid standard contradictions, but I must confess that I've never checked such arguments.


I don't know many references for this question. Lurie discusses some constructions in section 5.4 of [Higher topos theory](http://arxiv.org/abs/math.CT/0608040).


