Non-set theoretical mathematics indeed stay within the limits of the continuum (and rarely its power set, e.g. Lebesgue measurable sets is a collection of size $\beth_2$).
In set theoretical aspects the power set of the continuum is the number of ultrafilters on $\mathbb N$ (assuming the axiom of choice and whatnot); there are a few theorems discussing $\beth_2$ under additional axioms and CH; and when you decide to toss away the axiom of choice and take the axiom of determinacy instead you find yourself in a strange model in which every countable partially ordered set can be embedded into cardinalities below the continuum.
It is also not true at all that there is no talk about cardinalities above $\beth_2$. It appears in PCF theory (and the PCF theorem itself), large cardinals are usually strong limit cardinals, so they dwarf the continuum much like the universe dwarfs an electron, and of course the choice-less contexts often throw a fit and head out to a whole other direction with cardinals (i.e. cardinals which cannot even be compared with the continuum).
In the rest of mathematics, the concrete object $\mathcal P(\mathbb R)$ forms a Boolean-algebra, it does not have a natural linear ordering, and it is mainly used as an example of an underlying set for structures of that size. However such objects are not very common (yet, anyway) since they tend to be a little bit too big to handle once you add structure which was not natural (in the sense that the real numbers have).
The more structure you want, the harder it is to handle larger and larger objects. Once you go beyond $\beth_1$ you cannot have both Hausdorff and separable topologies. One of my teachers once explained this very question to me with the answer that we can grasp finite things, and we can approximate countable (and thus separable) things. However beyond that it becomes very hard to work with things. There are objects which are very large, in modern fields such as C*-algebras you get to meet them from time to time, and slowly in other fields. However it is still convenient to work with separable/countably generated/finitely generated objects for most people. If you wait a century or two then I'm certain that larger constructions will seep through the cracks and become mundane.