After one of my many thought-provoking LRC articles on matters divine, an emailer thought he'd blow me up by asking, "Can God make a chair so heavy he can't lift it?"
I forwarded the email to a Christian philosophy professor at my college, and he said that most theist thinkers deal with this type of thing by saying that even an omnipotent being needn't have the ability to perform nonsense tasks. E.g. nobody would challenge Aquinas by asking, "Can your God blumber dift?" So in the same way, my colleague said, we shouldn't squirm at questions like this.
But I think in this particular case, that's not the right answer. It is certainly coherent to ask if God can make a chair so heavy He Himself can't lift it. Moreover, it's not even an internally problematic statement--I can make a chair so heavy that I can't lift it. (I concede that I can't make a chair so heavy that God can't lift it.)
Thus I think the correct answer to my smug critic is this: Yes, God does possess the ability to make a chair (or rock) so heavy that He Himself can't lift it, but He chooses not to. It seems to me that this response in no way infringes on God's omnipotence, and indeed fulfills it (whereas my colleague's response doesn't).
Finally, if you say, "OK, but can God make a chair so heavy that He Himself can't lift it, and then does He have the power to lift it?" at that point I would invoke the philosopher's response. I.e. here (unlike the first question), you really are asking, "Can God make A and not-A true at the same time, in a way that appeals to my intuition?" And no, you can't ask God to do the logically impossible, just like He can't be expected to blumber dift.