How big can a set be? I was reading that the ordinal numbers do not form a set because there are too many of them, instead they form a proper class. Is there a maximum cardinality for a set?

As the other answers have pointed out, there is no largest set.
However, there is a sense in which the answer to your question is "yes":

Suppose $C$ is any class. Then $C$ is a proper class (i.e., not a set) if and only if there is a (class) surjection from $C$ onto the ordinals.

In a sense, this says that the size of $\{$ordinals$\}$ is the smallest proper class "cardinality", analogously to how - while there is no "largest" finite set - $\aleph_0$ is the smallest infinite cardinality. (My use of "the" here is of course not quite correct, but oh well.) On the other hand:

It is consistent that there is no (class) surjection from the ordinals onto $V$.

So there can be different "sizes" of proper classes, and in particular the first statement above is non-trivial.
