How has the ontological argument been used to explain the Trinity?

The Ontological Argument

First, the ontological argument focuses on proving the existence of God through reason(source). The proponent of this argument was Anselm of Canterbury (1033-1109 C.E.).

In a nutshell, this logical proof for theism says that the perfect being you can possibly imagine is none other than God.

The other way to understand this is to mirror ourselves in God based on our God-given limited capacities. That is, try to think of God's image in us, those things which he imparted to us by creating us in order to bear his image and be like him.

With it we bless our Lord and Father, and with it we curse people who are made in the likeness of God. James 3:9 (ESV)

Imagine that our innate capacities are unlimited, that will make us conceive of a being greater than what we are.

Every good thing we do in our limited human nature is already beneficial for the whole. Just think how much more if we are unlimited? That would make our ontological existence best of the best.

Now this is seen in the light of God's εἰκών (Genesis 1:26-27 cf. James 3:9). However, being a phiolosophical construct per se, the ontological argument could stand without any preconceived ideas to support it. That is, the ontological argument could stand on its own.

How has the ontological argument been used to explain the Trinity?

The ontological argument is not about a specific deity. This a priori knowledge supports theism (the belief that there is God) per se regardless of who or what that divinity is.

The Ontological Argument and the Trinity Is the ontological argument compatible with the Trinity?

“Is Anselm's ontological argument compatible with the Doctrine of the Trinity?

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 (source).

Therefore, the ontological argument cannot be used to prove the Trinity. The number of the Christian God in regards to personhood is 'three' but we humans cannot think of why God is three persons since we do not have either the experience or reality of being multiple persons and one being at the same time. We can only know God is three persons by knowing what the Holy Bible says about the Christian God.

Based on your comments, I understand that you are interested if any sort of ontological arguments have been made for the existence of God in a Trinity; and not whether "The Ontological Argument" - usually assumed to mean the medieval argument of Anselm of Canterbury for the existence of God - has been used to prove that God exists in a Trinity.

Oxford dictionaries defines ontology as "the branch of metaphysics dealing with the nature of being". An argument comprises a series of premises intended to support one or more stated conclusions.

With these definitions in mind, an argument for the existence of God as Trinity that perhaps comes the closest to being ontological is that made by Metropolitan Kallistos Ware in his book, The Orthodox Way, where he claims that God's existence as Trinity logically follows from God's nature as being personal and being love. "Both these notions," explains Met. Kallistos, "imply sharing and reciprocity":

First, a “person” is not at all the same as an “individual”. Isolated, self-dependent, none of us is an authentic person but merely an individual, a bare unit as recorded in the census. Egocentricity is the death of true personhood. Each becomes a real person only through entering into relation with other persons, through living for them and in them. There can be no man, so it has been rightly said, until there are at least two men in communication.

The same is true, secondly, of love. Love cannot exist in isolation, but presupposes the other. Self-love is the negation of love. As Charles Williams shows to such devastating effect in his novel Descent into Hell, self-love is hell; for, carried to its ultimate conclusion, self-love signifies the end of all joy and all meaning. Hell is not other people; hell is myself, cut off from others in self-centeredness.

(Kindle Locations 408-414)

"God is far better than the best we know in ourselves," he continues:

If the most precious element in our human life is the relationship of “I and Thou”, then we cannot but ascribe this same relationship, in some sense, to the eternal being of God himself. And that is precisely what the doctrine of the Holy Trinity means. At the very heart of the divine life, from all eternity God knows himself as “I and Thou” in a threefold way, and he rejoices continually in this knowledge. All, then, that is implied in our limited understanding of the human person and of human love, this we affirm also of God the Trinity, while adding that in him these things mean infinitely more than we can ever imagine.

(Kindle Locations 415-419)

I think Met. Kallistos makes a good philosophical argument for why God must exist in more than one person, but I am not sure we can close the argument that He exists in a Trinity from ontology alone. Why, for example, could God not be a Duality or a Quadrinity? He goes on to write that "the Trinity is not a philosophical theory but the living God whom we worship; and so there comes a point in our approach to the Trinity must give place to wordless prayer." Other knowledge we have of the Trinity comes from truth that has been revealed to us, and is probably not something that we could induce or deduce.