I think we need to be careful anytime we suggest that God cannot do something. When we're thinking about God, the last thing we'd want to so is "put God in a box" so to speak based on our own limited human understanding. As I noted in my previous post Thinking in the right direction:

God is not like us. We are like God. Not in the New Age sense of being gods or even “god-like”, but instead being made in God’s image so we reflect a portion (albeit sometimes a tiny portion) of His glory. If we were to say that God is like us, we would be anthropomorphizing God, making Him like us. We should try to think in the right direction (top-down rather than bottom-up) regarding God.

So, we should be hesitant about naming things that God cannot do. However, many philosophers like William Lane Craig would say that there are in fact things that God cannot do. They suggest that God doing something that contradicts Himself, which contradicts His own nature, is impossible. Also, many suggest that God would not do things that are nonsensical, or that are logically impossible.

It's interesting to note that the Bible does clearly list certain things that God cannot do. For example, it says that "God cannot be tempted by evil" (James 1:13) and "it is impossible for God to lie" (Hebrews 6:18). So there does seem to be some things that God cannot do.

This, I'll suggest, is not because there is any limit to His power, but rather because God is not just powerful, but also possesses other attributes (like goodness, or love) which won't contradict one-another. For God to do contradictory things would essentially mean that God would do things that are nonsense, and a God who would do things that are nonsensical would be a lesser God, not a greater one.