Is there a set theory that handles collections of proper classes? Is there a class of all classes? If not, can I define the “banana” so we have the “banana” of all classes? Is there a banana of all bananas? If I define an apple of all bananas, is there an apple of all apples. Can this go on indefinitely? Does mathematics make any sense doing this?
If I take the limit of such a process, do I get everything, ever?
Edit: If you are going to downvote, at least explain why. This is a legitimate question.
 A: There is another option, which is to work internally rather than externally. You are imaginging that you begin with sets, and then form classes, and then form classes of classes, etc. The challenge, as Henning Makholm points out, is that it is hard to talk about classes "from below" in this way. You end up with a type theory on top ov $V$, and it is not clear how far you can extend the "class ordinals" beyond the ordinals of $V$. For example, which ordinals "beyond $V$" can be defined from below? 
But remember that the intended models of set theory with classes are of the form $V_{\kappa + 1}$, where $V_\kappa$ is a full transitive model of ZFC. 
So, work internally instead. Begin with a model $M$ of ZFC + "there is a $\kappa$ such that $V_\kappa$ satisfies ZFC". Then, change your viewpoint. Consider the elements of $V_\kappa$ to be "sets", the elements of $V_{\kappa + 1}$ to be "classes", the elements of $V_{\kappa + 2}$ to be "classes of classes", etc. Now you have a perfectly consistent system that allows you to study a wide collection of "transfinite classes" -- the model $M$ guarantees that everything is consistent. 
The difference between the two approaches is that, in the first approach, you are pretending to live in $V_\kappa$ and are trying to look "up" at "external" classes. In the second approach, you begin with $M$, so that you look "down" at "internal" classes.  
A: There's nothing that stops you from making a set theory which contains collections of proper classes (which you may call "bananas" if you so choose) and collections of those collections ("apples") and so forth, to any finite depth.
NBG set theory (or, even better, Morse-Kelley) can be thought of as being the first level of this, giving a formal existence to classes, which would allow us to speak about bananas in the same way we speak of proper classes while working in ZFC.
However, you're not going to get a fruit of all fruits at any level in the hierarchy, at least not if your fruits are going to satisfy some kind of comprehension axiom (without which they would be of dubious utility). This is because fruit-of-all-fruits plus comprehension would immediately lead to Russell's paradox.
Except for the unavailability of a fruit of all fruits, why don't we usually see this carried out in practice? Mostly because it doesn't really buy us anything that is worth the aesthetic cost, and there are better ways to do it.
Doing it straightforwardly, you would need a new logical sort (or some single-sorted simulacrum thereof) for each kind of fruit you work with -- essentially you'd end up reinventing type theory as a separate "class layer" on top of ZFC. It feels unaesthetic and unmotivated to splice together two different plans for avoiding the paradoxes of naive set theory in this way.
You would get much the same, without the PR baggage of type theory, by interpreting the usual ZFC axioms in higher-order logic, which would give you higher-order classes for free as part of the logical machinery. Again, however, it would raise the question of why have Zermeloan bottom layer at all if you're happy doing things in higher-order logic (with its lack of completeness and so forth).
The preferred alternative to both of these seems to be to work in ZFC + "there is an inaccessible cardinal $\kappa$", which automatically provides a transfinite fruit hierarchy above $V_\kappa$ with no additional technical apparatus beyond the existence of $\kappa$. This is much more convenient. In principle it is slightly more risky than assuming ZFC (whereas a finite fruit hierarchy with appropriate axioms would seem to be less of a risk), but it doesn't look like anyone seriously thinks ZFC+Inaccessible is really any less consistent than ZFC.
