What can't be described by categories? 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.
 A: 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.
A: There's an interesting variant of your question, which may perhaps have been included in the first part of it, as to whether there are parts of mathematics where categories have little traction, and whether this is a necessary state of affairs. I was driving at this distinction when I posed a question to Terry Tao which he answered here.
A: I don't know if this is the kind of answer you're looking for, but category-theoretic language cannot be used to define some properties you may be very interested in. For example, if you're working in a category where the objects have underlying sets, then the notion of a morphism being surjective is not a categorical one. You can prove that in the category of sets, surjectivity is equivalent to the map being an epimorphism, but this is not true in the category of topological spaces (where epimorphism corresponds to the image being dense).
So you can't do everything with abstract nonsense. Sometimes, you actually have to prove things by using special facts about the specific category you're working with. But those results can often be interpreted as "rules about how things work in the category". For example, you might prove that if two morphisms have a certain property P, then when you compose them, you still get something with property P. Or maybe you can show that "property P is stable under base change" (a common one in algebraic geometry). To prove these things, you have to "get your hands dirty," but once you have those results, you can use them in your category-theoretic arguments.
A: From the point of view of someone who uses mainly analytic inequalities applied to areas of geometric analysis and PDE's, the more natural question is "why is category theory useful at all?"
Category theory appears to be a resounding success in the more algebraic areas of mathematics, but has had far less impact in most areas of analysis and differential geometry. Why this is so I will leave to people who understand this much better than me.
A: Well, if you don't impose additional conditions, then of course even the category of all sets does not exist. To define such a category you either have to restrict to, say, sets of bounded rank or you have to take a set theory with a notion of classes for example.
The next step is the category of categories or functor categories. Here the situation is even worse (but similar). There are a lot of constraints imposed on category theory by set theory. Therefore people start to think about using categories and not set theory as foundation of mathematics. As far as I know, there is now adequate solution to these problems.
A: Category theory just provides a language. Sometimes it provides really useful descriptions and metaphors and theorems, but most theorems can't be proved using category theory (even if they can be nicely described in that language). The English language can be used to describe any mathematical object you can think of, and most proofs today are written in English, but that doesn't mean that somehow English has some deep mathematical significance.
A: Someone (Halmos?) said that category theory can only describe the most trivial parts of any subject, and that this a valuable service because it's good to clearly mark which parts are trivial.
