Peter van Inwagen in his recent book The Problem of Evil, claims that the concept of God is equivalent to the concept of the greatest (metaphysically) possible being. There is thus a flexibility in what attributes one ascribes to God depending upon what one thinks is metaphysically possible and what one thinks is a great-making feature.
This understanding of the concept of God seems to be too weak for several reasons.
First, suppose that very few things are in fact metaphysically possible. For instance, suppose that it is impossible to know any truths that are not necessary truths. It is at least epistemically possible that such things are metaphysically impossible. If that were the case, could God exist? On van Inwagen's account there could still be a greatest possible being, and thus God would be that being. However, if the best the greatest possible being could do knowledge wise was to know necessary truths, then I have a hard time seeing that that being is God. It seems to me that if there are significant limitations to what is metaphysically possible, then God could not exist. (whether there are such limitations is another story)
Second, suppose that an individual, Joe, has a diminished understanding of what is in fact metaphysically possible. So, though it is possible to know non-necessary truths, Joe does not believe that it is. Joe still believes that there is a greatest possible being, he just does not believe that that being knows contingent truths since he doesn't think that can be done. Does Joe have the right concept of God and just the wrong concept of what is possible? I don't think so. It seems that he also has the wrong concept of God.
To sum up, it seems like there is a 'ground floor' of what must be metaphysically possible, or thought to be metaphysically possible, if God is to exist, or be thought to exist. Anything below that 'ground floor' leaves too diminished a concept of the greatest possible being for it to be the concept of God. Thus, the concept of God does not just equal the concept of the greatest possible being simpliciter, more must be said about what is possible.