Are proper classes objects? 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?
 A: 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.
A: (Insert standard and obvious disclaimers about opinion vs. fact.)
Of course proper classes are mathematical objects.  The fact that we can say things like "the proper class M is a model of set theory" means that proper classes must be objects, if as you say "objects are the bearers of properties" in mathematics.
What proper classes are not, and the only thing they are not, is elements of a model of ZF.  They are elements of a model of NBG, which is just as good a first-order theory, and foundation of mathematics, as ZF is.  And they are elements of the ambient theory in which we speak about models of ZF.  (To be more precise, there are certain elements in a model of NBG called "proper classes" which correspond, in a certain precise way, to certain elements of the ambient theory of a model of ZF, which justifies their being given the same name.)
Regarding the latter point: all the time set-theorists prove theorems with hypotheses like "for any proper-class model of ZF, ..." or state the axiom of replacement as "for any class-function whose domain is a set, its range is also a set".  Of course such theorems and axioms cannot be formalized within ZF as single theorems, but they are stated in set theory books as single sentences (as they must be, since the set theory books have finite length)—and as such they are single theorems of mathematics about objects called "proper classes", despite their not being single theorems of ZF.
I'm guessing the insistence that proper classes are "not objects" stems from a belief that all of mathematics takes place in ZF.  I find such a claim much harder to justify than the slightly different claim that all of mathematics could be coded into ZF, were anyone so inclined.  It could also, of course, be coded into many different foundational systems.
As to the question of how to distinguish between "well founded hereditarily extensional objects that are sets and those that aren't," I would say that a well-founded hereditarily extensional object that is an element of some model of ZF is a "set" according to that model, whereas one which isn't, isn't.  Obviously this varies with the model we choose (and only makes sense in some ambient context where we have "models" to speak about), but that's the way things are.  There is no Platonic universe of "real" mathematical objects that contains "all" the well-founded hereditarily extensional things, from which we could ask to separate out those that "are" sets from those that "aren't".
A: Let me offer another answer in counterpoint to Andreas's answer, by pointing out a number of cases in set theory where it seems that a second-order treatment of classes, as in Goedel-Bernays set theory, seems fruitful in contrast to the definable-classes-only approach.


*

*First, much of our understanding of large cardinals is based upon a felicitous use of large cardinal embeddings $j:V\to M$, which are all proper classes. And set-theorists routinely quantify over the meta-class collection of such embeddings. For example, a cardinal is measurable if it is the critical point of such an embedding; it is strong if for any $\theta$ there is such an embedding with $V_\theta\subset V$; it is supercompact if such embeddings can be found with $M^\theta\subset M$. Although in each case we have a first-order combinatorial equivalent to the large cardinal concept in terms of the existence of certain kinds of measures or extenders on certain sets, nevertheless it is the embedding characterizations that have a robust and strongly coherent power that unifies our understanding of the large cardinal concepts. This seems to be a case in which treating the embeddings as objects has deepened our knowledge. 

*The Kunen inconsistency result, the assertion that there is no nontrivial elementary embedding $j:V\to V$, becomes trivial when one treats all classes as definable. One can easily rule out all such definable $j$, if one only cares to consider the case in which $j$ is first-order definable with parameters, and one needs neither the axiom of choice nor any infinite combinatorics to do it. (Just argue like this: the question of whether a given formula $\varphi(x,y,p)$ with parameter parameter $p$ defines such a $j$ is a first-order property of $p$, and so one can define $\kappa$ to be the least possible critical point of such a $j$ arising from any $p$, and this contradicts the fact that $\kappa\lt j(\kappa)$, since $j(\kappa)$ would also be defined this way.) The various formalizations of the Kunen inconsistency is explained in the first part of our recent paper on Generalizations of the Kunen Inconsistency. Note that Kunen formalized his theorem in Kelly-Morse set theory, in order to have a way of expressing the elementarity of $j$, but it turns out to be possible to formalize this in GBC.

*Class forcing is vital to much of our understanding of the relative consistency of global assertions, such as the full class version of Easton's theorem or the fact that supercompact cardinals are relatively consistent with GCH and with V=HOD. But a development of class forcing is most direct in a context, such as Goedel-Bernays set theory, where classes can be treated as objects. For example, when forcing the GCH with class forcing, the generic class will not be definable in the forcing extension, so this is a case where in order to achieve the extension we most naturally want to consider non-definable classes. 
A: I think Andreas provided an excellent answer, and he pointed out that his answer expresses his philosophical opinion, not some absolute mathematical truth.
I wanted to add some things, though.
Russell has shown to us that unrestricted freedom to form "objects" leads to a contradiction.
What you have in mind seems to be some sort of type theory, as your "hyperclasses" have indices ("$\alpha$-hyperclasses") and only consist of objects of lower index.
This should avoid the Russell paradox.  However, I always thought that it is one of the great achievements of theories like ZFC that they got rid of the technical complications of type theory.
Of course, there are set theories where proper classes exist as objects, for example Neumann-Gödel-Bernays set theory (NGB).  NGB has the same consistency strength as ZFC, so in some sense it doesn't give anything new.
Finally, mathematics seems to work out just fine in ZFC. Even though proper classes are not objects, this doesn't seem to be a serious obstruction and we can do everything that is normally done in mathematics.
