Furthermore, if we tried to formulate a conception of the character of God that described him as being unable to do anything against his own nature, we would end up with a self-referent piece of fluff that neither described nor clarified anything about the actual nature of God. Besides, God has done and continues to do things we don't understand, or for which we don't have a full enough picture of to be able to say whether or not it went against his nature in the first place. There is a black box phenomenon at work here. We understand God's nature to the degree we do, not because we can dissect him and see for ourselves what he is, but because we hear his word and see his actions.
All of that not withstanding, there is at least one thing God could not do even if he wanted to: God could not make a replica of himself. If God could be made, even in replica, then God wouldn't be the unmade. The great I AM wouldn't be but would begin. The Creator would be but a creation. If God could be more than one in essence, the ones being considered are not the One. If something else could be made almighty, then the almighty would be so no longer. No, the best that God could do along this line is to make someone like himself, in his image, but not him in his power and perfections.
Which brings me to another thing God could not do even if he wanted to--preserve his image in a being made in it while determining that being's actions. If a being were made in God's image, that being would have to have freewill analogous to God's, or it would not be in his image. God is not under necessity nor are his actions determined, and neither could a creature in his image be thus confined in will. This is verified by the descriptions of Adam's freedom in the Garden. He had the freedom to do a thing or to not do it, and God "waited" observationally to see what Adam would do.
So, though God is the very perfection of all that he is, in power and in ability, there are a few things that God cannot do, even if he wanted to.