First, let me say that the point I want to make has nothing to do with whether existence is a predicate. That is a separate issue ably debated by a number of philosophers.
Instead, I want to talk about whether greatness can be considered an essential property of a thing as opposed to a speaker evaluation of a thing.
As many of you know, Anselm reasons that God must exist, since that which doesn't exist would not be as great as that which does. Absent existence, God would not be that than which nothing greater can be conceived.
It seems to me that this argument depends on the notion that greatness can be an intrinsic property of something, including God. If greatness is only a speaker evaluation of God, rather than a property of God, then this evaluation is a property of the speaker. Since greatness itself cannot be intrinsic to anything as a matter of objective fact, then the ontological argument can't prove what it purports to.
Second, let's consider some of God purported attributes:
omnipotence: The ability to do anything. (I do not believe that it should be necessary to qualify this with phrases as in "anything logically possible," since logical impossibilities are not actual things that no agent can do, but instead meaningless verbiage. "Round square" is not a shape that can't be drawn. The phrase is instead meaningless verbiage a set of points that both are and are not equidistant from some center point.)
omniscience: The knowledge of every state of affairs. (Again, it should be unnecessary to qualify this with phrases as in "every extant state of affairs," since there are no states of affairs that don't exist, either physically, mentally, or both.)
incorporeality: Lacking a physical body. (I don't believe that there are good reasons to dismiss 'incorporeality" as a self-contradictory expression. As others have stated, supposing this incoherence makes a needless mystery out of why the term "incorporeal spirit" is used and understood cross-culturally around the world. What is more, gravity does not have a body. Neither do the other known forces. Yet it would be silly to assume that these forces cannot cause events.)
For my money, these supposed attributes of God are clear, if abstract. Contrast them with greatness:
Third, let's take a look at greatness: Patton was an unusually competent warrior, so he is called great. Mohandas Gandhi made influential use of non-violent resistance to undermine an oppressive colonial power, so he is called great. Vladimir Horowitz played the piano exceptionally well, so he is called great. Although the Great Wall of China did not stop invaders as it was meant to, it is very big, so it is called great.
The only relevant property that any of these "great" things have in common is that people exalt them, each for a different reason, i.e., on account of different properties that these "great" things have.
Finally, let's take a look at God's greatness. We might say that God is great because of his omnipotence, but anyone who has contemplated the problems of Evil and Natural Suffering might say otherwise. We might say that God is great because of his omniscience, but since God does not share arbitrarily large amounts of his knowledge with human beings, what's so great about his being a know-it-all? As for incorporeality, what the heck is so great about that? Empty space is incorporeal by definition. Yet almost no one exalts it, worships it, or calls it great.
tl;dr: Anselm's version of the ontological argument doesn't work because greatness is not an inherent or defining property of anything, but is instead a speaker evaluation of any number of collocations of properties, which evaluations will vary across contexts, speakers, and social groups. God doesn't exist necessarily just because some folks think he's great.