Let us suppose that we have various "axioms" that we hold to be bedrock, how-could-you-think-otherwise, truths, but we also want to have as few reasons as possible, perhaps due to Ockham's Razor. So, we may either take one of those axioms and try to derive all others from it, or we may posit a new axiom that ought to explain all other axioms, let this be Axiom Zero.
Well, what are the axioms we want to explain? There is the principle of non-contradiction, which says that something cannot be both true and false at the same time in the same way. Any number of things can be said to stand in some relationship to each other. These are just logical axioms, tons of math and logic can be reduced to this. "I exist" may also be held to be an axiom. There are other beliefs that we hold axiomatically, they just are not able to placed in the same way, and many of them can and have been questioned. Consider: The world exists, and is made up of sensible matter, our senses are reliable, if I have memory of sometime, and can say how I have moved from then to now, then that point in time existed. People have shown that these "axioms" need not be held as axioms, but they tend to be believed axiomatically.
So, what can we say about Axiom Zero? It must contain in it at least the ideas of consistency, relation, and existence. If it is to be the source of those axioms, then it must hold those ideas in itself as a coherent whole that is not merely the conjunction of the parts, and it cannot hold any of those ideas to a lesser extent than the derived axioms hold them, though perhaps less clearly.
In order that it might hold all possible axioms, it must therefore hold all truth implicit in itself. Thus it holds the truth that x and y stand in a relation R to each other, and similar truths. Because this axiom holds the idea of existence in itself in its rawest form, it must entail the existence of things, everything since it holds all possible axioms, not merely the necessary axioms, and thus if it were all that was, it would mean that something else, in fact everything that does in fact exist, also existed. Thus axiom zero cannot exist alone. Not only this, but it must contain as implicit in itself its own existence.
I do not think that one could find an axiom that does all of this. However, God does do all of this.