There are many foundations to set theory, ZFC, NBG, SEAR, to name a few, and while they differ in how sets, classes, and higher-order collections are represented as mathematical objects, they all attempt to provide useful tools to mathematicians without being glaringly inconsistent.
Note that many set theories stop at the set, or only weakly partially define classes. But for those that go further, I have the following question:
What kinds of operations are well-defined when working with sets, classes, conglomerates, and yet higher order collections? Or, what kinds of operations are commonly assumed to be well-defined in fields such as category theory, etc.?
Here are some operations which I am curious about.
1) Applying the axiom of choice to the class of all sets, or to the conglomerate of all classes, and so on.
2) Given two classes $A$ and $B$, forming the set of classes $\{A,B\}$. Similarly for sets of conglomerates, etc. Of course, the set of all classes should be disallowed by Russell's paradox.
3) Given two classes $A$ and $B$ and a set $C$, make a "set-class function" mapping $C$ into $A$, a "class-class function" mapping $A$ into $B$, and a "class-set function" mapping $A$ into $C$. Similarly for any pair of higher-order collections of any type.
4) Regarding the cardinality of a set, class, conglomerate (etc) as a way to make equivalence classes of sets, classes, conglomerates (etc), respectively.
5) Regarding the cardinality of sets, classes, conglomerates, and higher-order collections as a way to define an equivalence class among all mathematical objects.
6) Forming the set of all "class-class functions" (or is it a class?)
I'm the least confident that #'s 5-6 will be acceptable, but the rest seem reasonable to my untrained eyes.
Best Answer
This question is considered in detail in Mike Shulman's Set Theory for Category Theory.
The "big picture" way to think about it is to think that each time you do a power-set-type operation, you are constructing an object of a "higher order". You just then need to postulate high enough orders to do whatever it is you want. $U_0$ is the class of all sets, $U_1$ is the class of all classes of sets, etc.
Once you are forming classes of classes, etc., you are strictly outside what is provably consistent with ZFC, but set theorists routinely consider much, much stronger theories that do not seem to harbor contradiction. So it's probably okay.