Omnipotence is a tricky thing. It seems so simple to say that something is all-powerful, but a close examination quickly reveals the flaws. What exactly does it mean to be all-powerful? Most people who use the term seem to think that it means there is nothing impossible for an omnipotent being. This, however, leads us into a realm of odd contradictions. Is it impossible for an omnipotent being to find something it cannot do?
Even ignoring clever wordplay, it is generally agreed that there are a few things that an omnipotent being cannot do and yet still maintain its claim to all-powerfulness. These fall into the category of logical impossibility: things whose very definition involves a contradiction such as a square circle. Following the line of Aquinas, most knowledgeable theologians assert that omnipotence, especially where god is concerned, does not cover logical impossibilities.
Other than that caveat, though, everything goes. Should we find anything else that an omnipotent being cannot do, we must seriously question whether such a being exists. Now it should be fairly clear that, when judged in conjunction with other typical divine attributes, we can conjure up all sorts of contradictions. But even on its own, omnipotence doesn't fare too well.
Practically everyone who has examined omnipotence is familiar with the classic paradox: Can God create a stone so heavy even he couldn't lift? The upshot of the question is of course that there is always at least one thing an omnipotent being cannot do. This conundrum has seriously vexed those who wish omnipotence upon their gods. Short of limiting omnipotence in some way, which they are wont to do, there is no escaping this paradox.
Still, those brave theists couldn't help but try. The paradox, they say, falls into the realm of logical impossibility, and so does not affect God's omnipotence. This, however, seems to be a matter of definition in reverse: the paradox is claimed logically impossible because God cannot satisfy it.
An example may prove helpful. A few weeks ago, I helped some friends move. One of them is an excellent packer. As a consequence of this she can, and did, pack boxes so heavy that she could not lift them. On the other hand, she had the good sense to hire a moving company and the strapping lads weren't as efficient as she was. They could not pack a box heavier than they could lift.
Here we have the perfect illustration that the question is not logically impossible. We have two people, one who can create a box heavier than they can lift, and one who cannot. It is only when someone claims to be able to do both that we encounter a problem. Normally, we'd call this person a liar. But in the case of a being whose very definition demands they be able to do this, we are forced to conclude that it does not exist.