GeekPress points to a rather long list that purports to summarize the many attempts to prove that God exists. Of course, the list is flippant and humorous, but the point is fair. Can it be proven that God exists?
Well sure, but as I've written before, not by us. If God exists and wanted to prove his existence he could certainly do it, but he hasn't done so. Therefore, either he doesn't exist, or he doesn't want to prove it. Which is it?
Hebrews 11:1-2,6 -- Now faith is being sure of what we hope for and certain of what we do not see. This is what the ancients were commended for. ... And without faith it is impossible to please God, because anyone who comes to him must believe that he exists and that he rewards those who earnestly seek him.
It might seem like a convenient out, but the fact of the matter is that God requires us to come to him by faith. There is quite a bit of evidence that God is real, but that same evidence can be interpreted in many ways, and it doesn't constitute "proof". It's very unscientific, I know.
Why would God do this? I can only speculate, but it might have something to do with free will. If God wanted to compel us to behave in certain ways (to believe in him or not, or anything else) then he could do so, but instead he has endowed us with the power of self-determination. Consider: if there were incontrovertible proof that God existed, wouldn't that be nearly the same as compulsion?
Most of the non-Christians I know (even atheists) agree that if it could proven to them that God exists and wants them to do such-and-such, they would do it. That may or may not be the case (consider those who met Jesus face-to-face and still rejected him), but they agree rationally that when confronted with proof, acceptance is the only reasonable action. However, God doesn't want to force us into choosing him, he wants us to love him as a response to his love for us.
This line of thinking is one possible component of the truth, and there are many others, but in the end it comes down to free will. God wants us to choose him out of our own free will, not because he makes the decision inescapable.