I'm not saying it's not interesting—in fact, I think it's fascinating—but all of this is implicit in Cantor's Theorem. "Harder question" to me implies there's something there that goes beyond the fundamental result that the powerset of a set X has strictly larger cardinality than X. One might think that it's just harder to understand, but I would dispute that too: if someone thinks they get Cantor's Theorem but has trouble grasping that there is no set of all sets, or largest cardinal number, it just shows that they don't really understand the theorem itself yet.

Like I said in my response to pndmnm, in my view if someone "[would] think that the cardinals also formed a set with an (infinite) cardinality of its own" then they haven't really grasped the theorem yet. It's built into it that if we're always allowed to form the powerset of a set then regardless of how large a cardinal you can find, you can always take its powerset and obtain a larger cardinal.