As many other have just said you cannot think to study just some particular subjects ignoring some other areas, expecially if you want to do research.
Most of math was born from the observation of some similar phenomena in many different areas: for instance the concept of category itself was born from the observation that in math we deal every time with collections of structures and morphisms preserving those structures, that led to the abstraction of category, similarly I strongly doubt that Grothendieck could invent the concept of (generalized) sheaf if first he hadn't known the many concrete sheaves that appear in topology, differential geometry and algebraic geometry, so it couldn't get to the concept of (Grothendieck's) topos, and without that I'm not so sure that Lawvere could get to the concept of elementary topos while doing his research in logic. This are just some example of as math have evolved thanks to interaction of different areas (for instance, as you can see in the example above, from interaction of geometry and logic).

Just to answer to your comment about analysis there's a professor in Italy who studies higher dimensional category theory for his research in analysis, so analysis need higher category theory.

Of course the best place where you can get a lot of intuition of higher category theory is algebraic topology where higher categories are used to model homotopy types for topological spaces, via $\infty$-groupoids, and directed space, via $(n,r)$-categories where $n,r \in \omega \cup \{\infty\}$ but you can find a lot of higher dimensional category theory in logic and computer science too, I've seen some application in calculability theory and model theory where (higher) category theory is used to model the semantic of theories, in particular type theory (if you're interested in application of higher categorical logic-model theory you can take a look to Makkai's work and also Mike Shulman's work on homotopy type theory). Also in mathematical physics there are a lot of higher category theory as John Baez's work prove.

I suppose above you were referring to Cheng-Lauda "Illustrated guide book", that's a good book if you want to learn many approaches to $n$-categories, but in higher category theory there's a lot of more then just $(n,r)$-categories (like usually Mr.Shulman says), Leinster's "Higher operads, Higher categories" is more complete from this point of view because it presents a lot of stuff like generalized multicategories/operads or $fc$-multicategories. Anyway if you want some references on higher category theory you can find some here.

Hope this may help you.

(Edit: I've improved a little the answer now that I've found some other references.)

really lovemaths, then you may underestimate how much a PhD demands of you. – David Roberts Sep 18 '11 at 22:19