I would go so far as to say one has to not think of things larger than the collection of classes (in so far as such a thing is even defined) not as just 'bigger sets' but something else. I posit that that something else is ... categories! In particular, not just categories in the fairly vanilla sense as a class of objects (which may be a set) and a set of arrows (or perhaps a class) between any two pairs of objects, but using the first order definition of a category _without an equality predicate on objects_, and a dependent-type version of equality on objects (can only compare arrows if they are in the same hom-collection). 

A small category with an equality predicate on its objects admits a(n essentially) surjective functor from a discrete category if we assume enough choice. In ordinary foundations (such as ZF(C), NBG or variants), Vopěnka's principle is a large cardinal axiom equivalent to the assertion that there are no subcategories of a locally presentable category (e.g. $Set$) which are simultaneously large (have a class of objects) and discrete. The principle can be seen a shadow in ordinary foundations of the idea that there should be categories which are really just _too big_ to have a collection of objects that behaves like a set. Notice that one can  form the posetal coreflection $Pos(C)$ of a category $C$ (it has the same objects and there is a unique arrow in $Pos(C)$ between any two objects if and only if there is any arrow between the analogous objects in $C$), and even take the core (the largest subgroupoid) of this. But we cannot get a category with an equality predicate on its objects unless we are happy to form some sort of quotient of $Core(Pos(C))$ to get a discrete category, and then it requires serious use of global choice on these super-large 'collections' to turn the canonical functor $Core(C)\to Core(Pos(C))/\sim$ into a functor $Core(Pos(C))/\sim \to Core(C) \to C$ to get an essentially surjective functor from a discrete category.

As a sort of half-way between this notion of category which is too large to be a class, we have the first-order characterisation of the category of classes, otherwise known as [algebraic set theory](http://www.phil.cmu.edu/projects/ast/index.html). One could apply the more philosophical ideas from the above paragraphs to the very concrete definitions of algebraic set theory (see for instance section 3.1 of [this introduction](http://www.staff.science.uu.nl/~berg0002/papers/pamphlet.pdf)). One would then have a category of classes which is itself genuinely not a meta-class, nor some sort of collection which behaves like a class, nor just a 'class' corresponding to a large cardinal in model/universe of set theory containing the current model/universe. This would probably require playing around with the axiom (US) (section 3.1 [here](http://www.staff.science.uu.nl/~berg0002/papers/pamphlet.pdf)).

Mike Shulman has some good comments on similar (though less extreme) ideas in [this answer](https://mathoverflow.net/questions/29302/reasons-to-believe-vopenkas-principle-huge-cardinals-are-consistent/29473#29473).

If one complains that this is just a first step, and really we want a whole hierarchy  of notions of 'bigger than anything we can come up with so far', then Michael Makkai has considered foundations (a sort of type theory, called by him [FOLDS](http://ncatlab.org/nlab/show/FOLDS)) in which it is impossible to consider equality (as above), isomorphism (as might be considered natural in 2-categories, for example), equivalence,... so that we can really only talk about each of these notions if we are working in an $n$-category for some finite $n$, and in general the only available generalised notion of equivalence is full-blown $\omega$-equivalence of $\omega$-categories. But this sort of approach has not been thought of in the sense of making larger and larger hierarchies of objects. (It has come up in Voevodsky's univalent foundations, but only from a homotopy point of view.)