I was recently asked to “prove that it is even possible for God to exist.” In order to answer this challenge, we need to define some key terms. I will leave off “prove” for reasons that I think will become clear.
When I speak of God, I am referring to a being that is all knowing, all powerful, everywhere present, unchanging, good, rational, wise, and loving, and holds all these attributes perfectly and without limit. He is also self-existent, which means he is not in any way dependent on anything or anyone else for his existence, rather all else that exists is dependent on God.
With respect to time, I hold that he is timeless without creation, but temporal since creation. This is important to note in order to explain how it is more reasonable to think God is the one who brought the universe, all of matter, energy, space, and time, out of nothing. It is my view that time is simply the relation of before/after, duration and interval between events, where an event is a change in the state of affairs. On this view, there was a state of affairs where all that existed was God. God created the universe, and with it time. He has since sustained the universe for some length of time (it is beyond the scope of this post to argue for how long that has been.)
So how does this prove that God is possible? Now we have to define possible. Philosophers divide possibilities into three modalities: logical, metaphysical, and physical.
“…on the standard model of the relation between these kinds of modalities the logical possibilities are the most inclusive; they include any proposition that sheer logic leaves open, no matter how otherwise impossible it might be. The metaphysical possibilities are the logical possibilities that are also allowed by the natures of all of the things that could have existed. The physical possibilities are the logical and metaphysical possibilities that are also allowed by the physical laws of nature. 
So rather than “prove” it is possible that God exists, I need only show that his existence is consistent with at least one of the above modes of possibilities.
Is there any law of logic violated by God’s existence, or even the proposition “God exists?” It is not an identity statement, so there is no violation of the Law of Identity. There is no compound proposition from which an excluded middle could be suggested, so there is no violation of the Law of the Excluded Middle. And, since there is no claim that God exists and does not exist at the same time in the same sense, there is no violation of the Law of Noncontradiction. So in this case, asking for proof of logical possibility is really asking for proof of a negative, and really the burden should be on the one who thinks it is logically impossible since it would be so easy to meet it, but that’s just my opinion.
Is God’s existence metaphysically possible? Metaphysics is the study of things and what kind of things they are. In a sense it is the study of what is and what can be. It seems to me that a being with the attributes listed above is the kind of being that is among those things that could have existed. There is nothing about such a being that is incoherent. For this distinction, however, let me illustrate the difference between metaphysical and logical possibility. It is strictly logically possible that the Prime Minister is a prime number (there is no violation of the laws of logic.) However, since prime numbers are not the kind of things that by nature are Prime Ministers, it is not metaphysically possible. To say that God is the creator and sustainer of the universe is logically possible, and metaphysically possible since the kind of being God would be is the kind of being that could create and sustain the universe. Conversely, it is not metaphysically possible that God is the Flying Spaghetti Monster since the FSM is a material being and material beings are not the kind of things that can exist timelessly and unchanging.
With respect to physical possibility, there is nothing in the laws of nature that precludes the existence of God. This is a separate issue from whether it is possible to empirically detect God. How we can even in principle know God exists is a distinct issue from whether it is physically possible. It may even be fair to say that to ask the question of physical possibility is a category error since God is not a physical being. However, something is possible just in case there are no impossibilities against it.
So in all three modalities, it is possible that God exists. Ordinarily, I think whoever is making a claim bears the burden, and I have tried to support my claim that God’s existence is possible. However, it seems that taking a stance that it is impossible is to hold that the idea violates logic, or God cannot be the kind of thing that could have existed, or that there is a law of nature that precludes such existence. I would love to hear which of these is the case with respect to God.