If God exists, would God create anything at all? The problem of non-God objects (PONGO) has to do with the fact that anything exists at all besides the God of classical theism. In other words, if God exists, then only God should exist; God wouldn’t create anything.
From Problem to Argument
Obviously, this alleged problem can be turned into an argument (and indeed it has). Here is the argument:
1. Necessarily, if God exists, then God-World (a world where God alone exists) is the unique best possible world.
2. Necessarily, if God-World is the unique best possible world, then God would maintain God-World
3. God-World is false because the universe (or any non-God object) exists.
4. Therefore, God does not exist. 
This argument is a bold argument because it’s a deductive argument, which makes it easier to escape. Thus, I propose a new argument that is more inductive (or evidential) in nature.
Towards a New Argument From “Non-God Objects”
If naturalism is true, there is a (natural) reality with no God. If classical theism is true, then God exists; however, it is an open question whether God would create anything at all. Thus, naturalism entails that there will be a part of reality that is not Divine, whereas theism doesn’t entail that there will be anything other than God.  Thus, the fact that there exist some entities other than God (if God exists) is evidence against God’s existence.
1. It is a known fact that reality consists of some things that are not God
2. (1) is more expected on the hypothesis of metaphysical naturalism than on the hypothesis of classical theism
3. The intrinsic probability of metaphysical naturalism is equal to that of classical theism
4. Therefore, other evidence held equal, classical theism is probably false
This argument isn’t saying that God would create other gods, whether those gods be limited or ‘unlimited’. Rather, the argument is concerned with the fact that there is a part of reality that is not the God of classical theism. In other words, there exist some things which are not God (e.g. beds, cars, flowers, universes, etc). Fundamentally, the issue is not whether God would create other gods; instead, the issue is that there exists anything at all apart from God.
I suspect the second premise is the most controversial. One might object to this premise by claiming that God has to create something outside of himself. I don’t (nor do most people) find this view of God and creation to be plausible, but let’s put that aside.
The issue now is whether that view of God and creation is more plausible than naturalism in explaining the fact in question (i.e. some reality not being God). We know that naturalism entails the fact in question, but we don’t know that said theistic view entails the fact in question. Sure, it might even be a reasonable belief, but the view does not amount to knowledge. Thus, our confidence level in naturalism should be greater than our confidence in this particular version of theism in explaining the fact in question. In sum, I won’t say anything more about this objection because this view of God is not popular.
To be sure, there is a lot more that can be said about the argument I’ve given. For example, I’m sure there are more objections that one can raise against it.
 This is Justin Schieber’s argument: https://freethoughtblogs.com/reasonabledoubts/2012/08/29/rd-extra-the-problem-of-non-god-objects/
 See my previous posts for a definition of classical theism (and metaphysical naturalism).