Hashem can do whatever He feels like doing. Nothing gets in his way.
This has led to the various questions on here regarding the omnipotence paradox - can Hashem put Himself in a situation in which He's unable to do something (i.e. create a rock He can't lift, create a prison He can't escape, destroy Himself, etc.)?
There are essentially two lines of thought, it seems, to answer these questions, as described in the above links: either Hashem cannot violate logic or is above it.
Where is this all leading? This discussion. The one in which everyone vehemently replied no to the question of "Can G-d ever take on a physical form." Sure, the question was based on a faulty interpretation of certain pesukim, but it's a valid question, nonetheless. And the answer is that He cannot take on a physical form.
Why? Didn't we just get through saying He can do whatever He wants? As @DoubleAA pointed out, it wouldn't necessarily be a problem according to those who say He can't override logic, but the question still stands for everyone else. Before you say that it's just that He chooses not to, that discussion makes clear it's not that He won't but that He can't. Why not?
(Yes, I'm aware that there are those who hold He can become physical; I'm asking specifically based on the opinions cited in the above discussion.)