This chapter from Groothuis' Christian Apologetics on the ontological argument will probably boggle my mind forever. Here's what I've got, though.
The first premise below is okay, because the idea of a perfect being is conceivable (not contradictory).
Anselm's first ontological argument:
1. God is the greatest possible being.
2. Actual existence is better than possible (mental) existence.
3. Because 1 & 2, therefore God, as the greatest possible being, does not merely have possible (mental) existence, but also actual.
Question: What if the greatest actual being is not the greatest possible being?