“Is Anselm's ontological argument compatible with the Doctrine of the Trinity? I can conceive of a being with more than three persons, so does that mean God has more than three persons? I understand quantity does not equal quality, but I can conceive of a being with at least 4 of the highest quality persons. I'm a Christian who's just starting to get into apologetics, and I can't find an adequate answer to this question.”
Thanks so much for your e-mail! I think you will find apologetics--and philosophical apologetics in particular--very intellectually satisfying, so let me just say welcome!
Now on to your question: if the objection at the question's heart holds, then it would be against any argument that relied on a greatest conceivable being (or Maximally Great Being, MGB), not just Anselm's (so this would work against Plantinga's contemporary modal ontological argument also, for example). However, I don't think the objection holds. There are a number of ways I can approach this question. I'll go in order (over a couple) from least convincing to most convincing. In my opinion, any of these are adequate to dispel the notion.
First, Anselm's argument for an MGB does not, strictly speaking, directly get us the Christian God. Anselm certainly meant it that way, and I think upon reflection we would see the Christian God is identical to the MGB. But since it's not designed to discuss that aspect, it doesn't appear to be able to be used for it.
Second, the problem of cardinal numbers presents itself. So if it's a matter of conception, one must add another number. But then one can always add another number. So if 4 is good, 5 is better; if 5 is good, is better, and so on. So with conception, one can conceive a potentially-inifinite number of persons. But it gets worse. For this number of persons must actually be concrete and instantiated in reality. Thus, all of the attendant problems with an actually infinite number of things present themselves/.
Third, the conception that matters is of metaphysical greatness, not merely any property. God has the accidental property of creating this world (the way reality is); very few people argue that this property is inherently great-making. For a reminder, a great-making property is a property it is metaphysically better for a being to have than to lack. I don't see any reason why God could not have created some other type of world, or even no world at all beyond himself and maybe a few angels. The point is that it's not necesarily true that the number of persons in a being is itself a great-making property. That the being of God is multi-personal is a great-making property is evident in that such a being will need to display love even without creation. However, this says nothing about the number of persons, and I don't see any good reason to think that four persons is metaphysically greater than three. Numerical greatness does not equate to metaphysical greatness; if it did, God would have to do one more good act than he in fact did, no matter how many he did!