I wanted to give a minor (but important) point which I think is amiss in this page.
We can classify notions of how big is a collection. In an over-simplification we begin with set (sets exist) to class (classes are definable) to $2$-classes, and so on.
Note that we have to improve our theory's capacity to discuss these things. Namely, we can talk about some collections of things which exist, but not on all collections and certainly not on collections of collections ($2$-classes in ZFC).
Russell's paradox is not really about sets. It is about collections. It tells us that some collections are too big to exist. We can easily replace the "set of all sets which do not contain themselves" with "the collection of all $2$-classes which are not members of themselves", which will prove that this collection is a $3$-class.
This goes on and on, and it proves that there is always a "bigger notion of size". While that for itself is important, and I will get to it in a moment, one can think it through and realize that this is really just assuming there is a $2$-inaccessible cardinal, and saying that everything below the $\alpha$-th inaccessible is an $\alpha$-class (where $0$-class is a set, of course). Then one can continue, on and on and on, until one gets to Mahlo cardinals and so on (as Joel and Andreas have indicated).
For this reason, I believe, it is important to actually fix some background universe from which there is no escape. If we assume that this universe contains sets and those sets obey the axioms of ZFC then this universe is not a set, of course. This universe is the absolute infinite and there is no classes beyond it.
Of course we are free to choose for different proofs and needs different "degree absolute" and Number Two to accommodate it with. However this is like deciding to live on a certain planet, for a while, then choosing another planet. We still have to stay in our universe; or dimension; or so.
Let me finish with my philosophical bent (which I have to admit has not yet been fully baked yet): there is such incredible universe which are are not privy to understand or see in fullness (or even know whether or not its axioms include ZFC), inside this universe there is a plethora of smaller universes of all sort of theories (ZFC+large cardinals, for example) which we can skip between whenever we need them.
Being strongly agnostic, however, I do not mean this existence in a Platonist way. I mean, at least for now, inside my head.