How many different sizes of infinity are there?

We don't really talk about "infinities", instead we talk about "cardinalities". Cardinality of the a set is the mathematical way of saying how large it is. Of course infinity could easily just mean $\infty$ which is a formal symbol representing a point larger than all real numbers (but the notion can be transferred to other contexts as well). This is not the same sort of infinity as infinite cardinalities. Infinite cardinalities are a whole other beast, and they are related to set theory (as we measure the size of sets, not the length of an interval).

Cantor's theorem tells us that given a set there is always a set whose cardinality is larger. In particular given a set, its power set has a strictly larger cardinality. This means that there is no maximal size of infinity.

But this is not enough, right? There is no maximal natural numbers either, but there is only a "small amount" of those. As the many paradoxes tell us, the collection of all sets is not a set. It is a proper class, which is a fancy (and correct) way of saying that it is a collection which is too big to be a set, but we can still decide whether or not something is in that collection.

In a similar fashion we can show that the collection of all cardinalities is not a set either. If $X$ is a set of sets, $\bigcup X=\{y\mid\exists x\in X. y\in x\}$ is also a set, and its cardinality is not smaller than that of any $x\in X$. By Cantor's theorem we have that the power set of $\bigcup X$ has an even larger cardinality.

What the above paragraph show is that given a set of cardinals, we can always find a cardinal which is not only not in that set, but also larger than all of those in that set. Therefore the collection of possible cardinalities is not a set.

If $A$ is a set, then the power set $P(A)$ is a set of bigger cardinality. If $\{A_i\}_{i\in I}$ is a family of sets, then $P(\bigcup_{i\in I}A_i)$ is a set of bigger cardinality than any of the $A_i$. This allows us to define an infinite set $F(a)$ for each ordinal $a$ such that $a<b$ implies that $F(a)$ has smaller cardinality than $F(b)$. To do so, let

• $F(\emptyset)=\mathbb N$,
• $F(a)=P(F(b))$ if $a=b+1$ is the successor of $b$,
• $F(a)=P(\bigcup_{b<a} F(b))$ if $a$ is a limit ordinal

Now if a set $S$ were able to enumerate all infinite cardinalities, this would give us an injective map from the proper class of all ordinals into this set, which is absurd.

As the other answers have pointed out (at least within the framework of ZFC), the answer is "proper class many." Let me just point out that this isn't the end of the story.

We can ask whether two proper classes have the same size, by asking about the existence of a definable bijection, etc. In this sense the class of ordinals - or equivalently of cardinalities of well-orderable sets (=initial ordinals) - is actually small: every other proper class surjects onto it! Given a proper class $C$, consider the map $rk: C\rightarrow ON$ sending a set $x\in C$ to the unique $\alpha$ such that $x\in V_{\alpha+1}-V_\alpha$ - that is, its (von Neumann) rank. This is well-defined, and maps $C$ to a cofinal subclass $S$ of $ON$. Now we can "collapse" $S$ onto $ON$ by sending $\alpha\in S$ to $ot(\{\beta\in S: \beta<\alpha\})$; this is surjective. Composing these two maps gives a surjection from $C$ to $ON$.

Meanwhile we can construct models in which there is a proper class $C$ with no surjection from $ON$ (let alone an injection into $ON$): see Joel David Hamkins' answer to https://mathoverflow.net/questions/110799/does-zfc-prove-the-universe-is-linearly-orderable.

Finally, we could ask: can there be a proper class $C$ such that $ON$ does not inject into $C$? This is a subtler question, and class injections are weird things; but I believe the answer is yes via class forcing (I vaguely recall seeing this a long time ago, and it being relatively simple, but I can't remember the details).

The distinction between injection and surjection above might make you think, "But wait a minute! Don't we have the axiom of choice to simplify things?" Indeed we do, but the axiom of choice treats only sets, and at the level of classes "injects into" and "is surjected onto" are still distinct, a priori. We can have an "axiom of choice for classes" (called global choice), if we enlarge our language a bit to talk about classes, and this axiom implies that all proper classes have the same size.