One student of mine recently expressed astonishment that someone besides God cannot have an omniproperty in the strict sense of the word. He was under the impression that a person (S) could have the property of omnipotence (OP), for example, and still not be God. Only a being having every omniproperty would be God (in his estimation).
However, there is more than one reason why his idea of God needs to be adjusted. I mentioned some reasons to him, and they appeared to be sufficient for the time being. But the whole point of the ontological argument for God's existence is supposed to be that only a perfect being is maximally great, and only a perfect being (in the absolute sense of the word) is necessary (exists in all possible worlds), and compossibly instantiates all omniproperties.
Another way of viewing this issue logically is to say that only God could be all-knowing and maximally powerful. How could a creature ever have such properties? Nevertheless, to emphasize this point, instantiating all omniproperties seems to be a logical consequence of having one omniproperty. For a being cannot be omnipotent or omniscient without being maximally excellent, it seems.
Two other ways of handling my student's question is to reference transcendentals like "good" "beauty" and unity; God also has incommunicable properties that only God exemplifies. See http://www.theopedia.com/classification-of-the-attributes-of-god
Duns Scotus argues that God's chief attribute is infinity. So he evidently contends that all divine properties are really just an extension of divine infinity.