Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the "process" of repeatedly forming the powerset of what has already been generated and, when appropriate, forming the union of what preceded.
But the practice of set theorists belies this, since they tend—for instance, in the theories of inner models and large cardinal embeddings—to study classes that, on pain of contradicting the standard axioms, are never "generated" at any stage of this process. In particular, faced with the independence results, many set theorists suggest that each statement about sets—regardless of whether it be independent of the standard axioms, or indeed of whether it be formalizable in the first order language of set theory—is either true or false about the class $V$ of all objects formed by the above mentioned process. For them, set theory is the attempt to uncover the truth about $V$.
This tendency is at odds with what I said set theorists study, because proper classes, though well founded and hereditarily extensional, are not objects. I do not mean just that proper classes are not sets.
Rather, I suggest that no tenable distinction has been, nor can be, made between well founded hereditarily extensional objects that are sets, and those that aren't.
Of course, this philosophical claim cannot be proved.
Instead, I offer a persuasion that I hope will provoke you to enlighten me with your thoughts.
Suppose the distinction were made. Then in particular, $V$ is an object but not a set. Prima facie, it makes sense to speak of the powerclass of $V$—that is, the collection of all hereditarily well founded objects that can be formed as "combinations" of objects in $V$. This specification should raise no more suspicions than the standard description of the powerset operation; the burden is on him who wishes to say otherwise.
With the powerclass of $V$ in hand, we may consider the collection of all hereditarily well founded objects included in it, and so on, imitating the process that formed $V$ itself. Let $W$ be the "hyperclass" of all well founded hereditarily extensional objects formed by this new process. Since we can distinguish between well founded hereditarily extensional objects that are sets and those that aren't, we should be able to mirror the distinction here, putting on the one hand the proper hyperclasses and on the other the sets and classes.
Continuing in this fashion, distinguish between sets, classes, hyperclasses, $n$-hyperclasses, $\alpha$-hyperclasses, $\Omega$-hyperclasses, and so on for as long as you can draw indices from the ordinals, hyperordinals, and other transitive hereditarily extensional objects well ordered by membership, hypermembership, or whatever. It seems that this process will continue without end: we will never reach a stage where it does not make sense to form the collection of all well founded hereditarily extensional objects whose extensions have already been generated. We will never obtain an object consisting in everything that can be formed in this fashion.
For me, this undermines the supposed distinction between well founded hereditarily extensional objects that are sets, and those that aren't. Having assumed the distinction made, we were led to the conclusion of the preceding paragraph. But that is no better than the conclusion that proper classes, including $V$ itself, are not objects. Indeed, it is worse, for in arriving at it we relegated set theory to the study of just the first two strata of a much richer universe. Would it not have been better to admit at the outset that proper classes are not objects? If we did that, would set theory suffer? In particular, how would it affect the idea that each statement about sets is either true or false?
Best Answer
Proper classes are not objects. They do not exist. Talking about them is a convenient abbreviation for certain statements about sets. (For example, $V=L$ abbreviates "all sets are constructible.") If proper classes were objects, they should be included among the sets, and the cumulative hierarchy should, as was pointed out in the question, continue much farther, but in fact, it already continues arbitrarily far.
In particular, when I talk about statements being true in $V$, I mean simply that the quantified variables are to be interpreted as ranging over arbitrary sets. It is an unfortunate by-product of the set-theoretic formalization of semantics that many people believe that, in order to talk about variables ranging over arbitrary sets (or arbitrary widgets or whatever), we need an object, a set, that contains all the sets (or all the widgets or whatever). In fact, there is no such need unless we want to formalize this notion of truth within set theory. Anyone who wants to formalize within set theory the notion of "truth in $V$" is out of luck anyway, by Tarski's theorem on undefinability of truth.
Considerations like these are what prompt me to view ZFC together with additional axioms (such as the universe axiom of Grothendieck and Tarski) as a reasonable foundational system, in contrast to Morse-Kelley set theory.
A detailed explanation of how to use proper classes as abbreviations and how to unabbreviate statements involving them is given in an early chapter of Jensen's "Modelle der Mengenlehre". (The idea goes back at least to Quine, who used it not only for proper classes but even for sets, developing a way to understand talk about sets as being about "virtual sets" and avoiding any ontological commitment to sets.)
Finally, I should emphasize, just in case it's not obvious, that what I have written here is my (current) philosophical opinion, not by any stretch of the imagination mathematical fact.