I don't think so. (Congratulations, you've found something that isn't wrong with the ontological argument!)
1. You can define something to be "godlike" if nothing greater than it can be conceived, in which case the conclusion of the ontological argument (if it worked) would be that at least one godlike thing exists. That's probably enough for anyone who actually wants to use the argument; they'd probably say that "obviously" uniqueness is a kind of perfection, or something.
2. The nearest thing I can see to a Cantor-style paradox would be if somehow the totality of thinkable things were necessarily "greater" than any particular thing. But "greater", whatever it's meant to mean (the vagueness of the terms is one of the problems with the usual ontological argument) isn't the same as "bigger", and you could probably get away with arguing that the totality of all thinkable thoughts isn't so "great" because it involves inconsistencies, or something.
(Given some of the other arguments Anselm makes, which involve saying e.g. that something that causes greatness must itself be great, I think he would have had trouble making that last argument. But the ontological argument is so weak that I feel one ought to make all possible excuses for it :-). )
Doesn't this assume some kind of total ordering of thoughts and wouldn't that lead to a Cantor-style paradox?