A frequent point brought up in theist-atheist debates, for obvious reasons, is the beginning of the universe and along with that, the beginning of time.
Either time had a beginning, or time stretches back into infinity. Neither one makes sense to me.
There's an argument that time can't be infinite, which I disagree with. I don't think it's a logical impossibility, but intuitively, it makes no sense to me. I mean, it can't go forever. At the same time, time having a beginning makes even less intuitive sense. What came before the beginning of time? It may not be a valid question, but intuitively, it doesn't seem to make sense that it's an invalid question.
I think this is a situation where our savannah-evolved, monkey intuition fails us. It works very well with situations we deal with commonly, like where an object will hit if we throw it just so. But outside that realm, it simply doesn't work. Intuitively, time doesn't slow down, and length doesn't contract for an object moving quickly, because it's not perceptible at the speeds we move at.
I think the beginning of time is the thing. It doesn't seem intuitive because it's completely out of our experience. We can't rely on our intuition or our gut-instinct, because they can't handle this situation. Instead, we have to rely on science and math.