Anselm's ontological argument (AOA) for God's existence was first described in Anselm's Proslogion:
[I]s there, then, no such nature [as God], for the Fool has said in his heart that God does not exist? But surely when this very same Fool hears my words "something than which nothing greater can be thought," he understands what he hears ... So even the Fool is convinced that something than which nothing greater can be thought is at least in his understanding; for when he hears of this [being], he understands [what he hears], and whatever is understood is in the understanding. But surely that than which a greater cannot be thought cannot be only in the understanding. For if it were only in the understanding, it could be thought to exist also in reality—something which is greater [than existing only in the understanding]. Therefore, if that than which a greater cannot be thought were only in the understanding, then that than which a greater cannot be thought would be that than which a greater can be thought! But surely this [conclusion] is impossible. Hence, without doubt, something than which a greater cannot be thought exists both in the understanding and in reality.
Yet for several reasons, Anselm's ontological argument (AOA) is insufficient to show that God exists.
1. Anselm's ontological argument conflates God with the concept of God, only proving that the latter exists.
2. Existence is not a property, so it cannot be part of a thing's definition. If adding 'existing' to something's definition implies that it exists, then one can prove that anything exists.
3. Even if a maximally great being exists, it need not be a single entity since it could be many beings of equal greatness. It also need not be divine. Possibly, each human is maximally great.
4. Imagining a maximally evil being does not reduce AOA to absurdity as some objectors have claimed because, if goodness is identical to existence as Anselm and other medievals assumed, then a maximally evil being is maximally nonexistent.
5. If greatness is objective, then one must believe in objective greatness to believe in God via Anselm's ontological argument. And it seems that one must believe in God to believe in objective greatness according to the moral argument, making Anselm's argument circular. However, AOA and the moral argument both argue from the existence of greatness to God's existence.