So here is a simple argument. An argument is different than a proof. This is not a proof, but more like a thought experiment. Also, not everything in this argument aligns perfectly with my beliefs about God. It is an argument for argument's sake. So here goes. Answer each of these questions to yourself as honestly as possible.
1. There are an estimated 100 billion galaxies in the universe, each with an estimated 100 billion stars. What is the likelihood that somewhere out there, there is a being of such an advanced level of understanding and power that he/she/it has powers we would consider "Godlike"? That he/she/it can read our minds? That to he/she/it, distance is irrelevant, and that he/she/it can arrange situations to create a desired outcome? That he/she/it can even speak to our minds? To understand the strength of this argument, check out this. In sum, some artificial intelligence theorists suggest that once we are able to teach computers to improve their own intelligence through re-writing their own programming, artificial intelligence will explode. It will expand so quickly that it won't be long before these computers would be as much above humans in intelligence as we are above ants in intelligence. Their powers could truly be god-like. If such a thing is so easy to conceive and seems possible, why wouldn't it have already happened somewhere in the universe? Isn't it conceivable that we are already under the control of such a being, for whom distance could be irrelevant?
2. What is the likelihood that such a being would want to help us advance to a higher level of being? Based on what we know about ourselves (which is really our only point to extrapolate from), altruism and the desire to "make a difference" might continue up the chain to higher level beings. This is an assumption, but what better assumption could we make? If this being wants to help us improve, he/she/it is starting to sound like a benevolent, all-powerful God who could actually be involved in our lives.
3. What is the likelihood that such a being would want to help us in an "indirect" manner so as to avoid destabilizing our natural growth process as a species and as individuals? To understand this argument, think of the "Prime Directive" in Star Trek. Essentially, a higher level being simply appearing (or descending in a space ship in Star Trek's case) and telling the lower-level species what to do would rob those beings of the chance to have their own civilization, their own development process. Again, this is an assumption, but I think a reasonable one. This would explain why God doesn't just "prove himself" to everyone in a very physical way. It would also explain why there are so many religions: God may be speaking to chosen messengers (so as to maintain indirect contact), but mankind has a tendency to twist, take advantage of, and create counterfeit versions of any ideological teaching. This is just the reality of our human tendencies.
4. What is the likelihood that such a being would avoid detection by human attempts to prove/disprove he/she/its existence? What I mean by this is simply an extension of the previous point. A God-like being, trying to influence us in a moral, indirect manner would not want to be "proved." Such proof would be essentially the same thing as he/she/it revealing themselves directly. Such a being would be perfectly able to influence events without leaving a signature that would show up on our relatively crude scientific instruments. Such a being would be able to see such puny experiments coming well in advance.
6. What is the likelihood that such a being would only reveal themselves to sincere seekers? Sincerity, which we could define as a real intention to act on truth once it is learned, is by logical necessity an essential condition for real personal growth. If this being could see into our thoughts, why on earth would he/she/it reveal themselves to anyone who was not sincerely looking for that revelation? This would explain why religious seekers report all kinds of miracles and experiences in their lives, while those who are content in their secular lives don't report similar experiences. Many seculars conclude that the religious seekers are making these things up or are unable to see the difference between natural coincidence and divine intervention, but what if the difference is really that God intervenes visibly with some and not with others?
5. What is the likelihood that a God-like being would want to propagate its own kind (its species)? All we know about life indicates that this desire is deeply central to our psyches. Does the idea that God created man "in his own image" sound so strange with this in mind? No we have an even more Biblical sounding being on our hands.
My point is not to prove anything, but rather to show that belief in a God is not irrational, based on what we know of the universe and about life. It actually makes sense rationally, whether it is empirically true or not. The God I have called into possible existence through this post matches with the God that is taught in The Church of Jesus Christ of Latter-day Saints, to which I belong (although there are many important aspects of our belief in God that I couldn't cover here). Essentially, He is a morally perfect being who has gained all of the power and knowledge which it is possible to gain. He is perfectly able to keep any promise He makes, because of his power and knowledge. He was once a man like us, but is now an exalted man. He spends His time and finds His joy populating the universe with more of His children, and helping them to ascend to higher and higher levels of existence.
