It is argued in some circles that god is by necessity an atheist. For, in order to know something objectively, you must stand outside and apart from it. Otherwise, your knowledge of a thing is subjective. Yet, god is infinite, which means god does not stand outside and apart from anything (or else god would be limited, which is opposed to god being infinite). Therefore, since god cannot stand outside and apart from anything, god must not be objective about god, and consequently god must be an atheist about god's own existence. Is this argument conclusive? If so, why? If not why not?
I disagree with the premise that in order to know something objectively, you must stand outside and apart from it, and that this knowledge is subjective otherwise.
And actually, internal knowledge in the form of, "I think, therefore I am" is perhaps more objective than any external knowledge could be. When we make this statement, we're saying that it's an objective fact that we exist, regardless of the problem that we may or may not be able to prove this objective fact to others (if there are others). We may not be able to know our true nature, but we can know that we individually exist.
To answer the question, a god would be a theist, but might possibly have some doubts depending on what type of god it is.
1) A god is necessarily sapient, because if it is not, I can likely offer a better definition for this entity than the word "god".
2) If it is sapient, then cogito ergo sum applies to it. It knows it exists. The only remaining question, then, is can it know for sure that it is indeed god?
3) Unless god has some way of knowing something for sure that is unavailable to other beings, it may be unable to surely conclude that it is god. Even if it knows everything that it seeks to know, and is able to control everything it knows in any way that it sees fit, can it be sure that it is not a small part of a larger picture? Can it know for sure that what it experiences is truly everything that exists? Could there conceivably be separate universes with separate gods, unaware of each other, but having omnipotence and omniscience in their own realms? I can conceive of a thought experiment where I am placed into a machine that fools all of my tools for perception, and leads me to believe I am god by allowing me to know and control everything about my virtual environment.
4) Aseity or eternity might provide the ability to know for sure that it is god, but it gets very abstract at this point. If god is capable of self-origination, or has no beginning, it is potentially bending logic itself, and becomes inconceivable. Perhaps it can eliminate the possibility that realms exist that it is not aware of.
So basically, it's either a gnostic theist or an agnostic theist depending on its qualities.