What in Mathematics cannot be described within set theory? I have begun reading Patrick Suppes' book Axiomatic Set Theory. The first sentence in chapter 1 reads: "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
What are these rare exceptions? What in mathematics cannot be described within set theory?

There's an area called Category Theory. It deals with the notion of a category, consisting of a class of objects (which in general is $NOT$ a set and morphisms between these objects. 
For example, there is a category called $set$ in which the objects are all the sets in math, and its morphisms are maps between sets.
Google it up. Also "proper class".
There's also the notion of cardinal of a set. intuitively, given a set $X$, $card X$ denotes the "quantity" of elements of $X$. The class of all cardinals is not a set.
