From a wholly different angle, the argument uses the premisse that we can conceive this greatest conceivable being. It is very much the question whether we can conceive the greatest conceivable being. I think many theists would agree if I said that God cannot be completely conceived by us at all. There is also the question whether there even exists something like a 'greatest conceivable being'. There might well be for every conceivable being a greater one. Or there might be two beings such that neither is greater than the other, nor any being that is greater than both.
-- Andre Engels