I suppose the answer to the question of God's 'omnimax'-like qualities hinges on the definitions one employs to describe these qualities.
I didn't spend a lot of time referring back to different definitions of 'omnipotence', but I did take a look at a few different online dictionary entries to see the 'standard' or most-widely used definition of the word. I didn't really like any of 'em.
Of course, there will always be some theists that claim God can do absolutely anything. He can dunk better than Michael Jordan, paint better than Picasso, shoot a rifle more accurately than Alvin York and tell a joke that's funnier than Richard Pryor.
However, while God can perform the impossible, God cannot do the illogical and the absurd. God can defy the laws of gravity, I would suppose. Einstein says matter can't travel faster than the speed of light, correct? You brainy science-types help me out here. I'm thinking God could break, ignore or simply circumvent this and other pesky rules of physics, if He wanted to do so. And if I'm wrong about Einstein and the light speed rule thing, I would ask you science-types to please correct or ignore me. Even if I'm a little off on the technical aspects of my own argument, I hope you all see and understand what I'm trying to say. God can probably create and destroy matter as well as energy, but He can't tell a truthful lie or make a circular square.
In my estimation, which I'll concede could be woefully wrong, logic, unlike physical laws, is not a set of natural 'rules' by which our universe/nature operates. Logic, unlike the laws of science, does not confine God nor does it impose limits on his omnipotence. Logic is a medium by which we pinpoint and communicate truth, or so it seems to me. So, the old paradox about God and the 'rock so heavy He Himself can't lift it' does not demonstrate God's divine limits; it does indicate to me, however, that mankind's ability to understand and convey certain truths is somewhat confined by our own flawed and fallible instruments of personal communication.