Necessary vs Contingent: Necessary facts (such as numbers) are true in every possible world (e.g. there is no possible world where the number three does not exist). Contingent facts (such as people) depend on something else for their existence, so there is a possible world where you and I do not exist.
God’s Status: God, meaning a “maximally great being”, would by definition exist necessarily, not contingently. If God existed contingently, then he would have an external cause and thus that would be God, not him. Thus, if God exists, he is necessary and not contingent.
1. It is possible that a maximally great being exists
Support:: A priori warrant. God is not incoherent at first sight, like a square-circle or a married batchelor.
Objection: Parodies of the ontological argument attempt to show that it can be used to prove all kinds of absurd things, like a most perfect island or maximally excellent lion.
Response: An omniscient, omnipotent, all-good being has attributes that have an intrinsic maximum. In contrast, a most perfect island can always have more palm trees and tiki torches and thus could never be imagined or realized. A "greatest island" is logically incoherent. A maximally excellent lion would have to be necessary and exist in all possible worlds, and if that were the case it could hardly be called a lion anymore.
Objection: Why not postulate a quasi-maximal being instead of a maximal one? No reason why one is preferable to the other.
Response: A quasi-maximal being is not a necessary being. A maximal being, being omnipotent, has the ability to refrain from creating anything at all. So there are possible worlds where no quasi-maximal being was created.
Support: A posteriori warrant. The other arguments for God might provide such warrant. As for quasi vs maximal, simplicity would lead one to propose either the existence or non-existence of a maximal being as opposed to a quasi being, as zero or infinity are more simple terms than some arbitrary number (as would be the case with a quasi being).
Objection: The argument is question-begging, as premise #1 is the same as the conclusion.
Response: Natural theology presents a cumulative case for theism; each argument is not a link in a chain but instead is a link in a coat of chainmail.
2. If it is possible that a maximally great being exists, then a maximally great being exists in some possible world
Support: Per the concept of possible worlds (a way reality could be), if it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3. If a maximally great being exists in some possible world, then it exists in every possible world
Support: See God’s status, above. Necessary facts are true in all possible worlds, and contingent facts are true in only some possible worlds. A maximally excellent being would, by definition, not depend on something else for its existence and thus would be necessary, not contingent.
4. If a maximally great being exists in every possible world, then it exists in the actual world
Support: The actual world is one of the possible worlds.
5. If a maximally great being exists in the actual world, then a maximally great being exists
Support: Self explanatory.
6. Therefore, a maximally great being exists
Support: From 5.