I think the classical answer to this is to say that there are two types of things that exist: things that might or might not exist, and things that have to exist. As the Stanford Encyclopedia of Philosophy puts it,
It is commonly accepted that there are two sorts of existent entities: those that exist but could have failed to exist, and those that could not have failed to exist. Entities of the first sort are contingent beings; entities of the second sort are necessary beings.
According to the Christian tradition, God is supposed to be omnipotent, omniscient, and perfectly good. His existence is not contingent on any outside forces. If it were contingent, then those forces would have power over Him, and he would not be omnipotent. Hence, it is supposed that God's existence is necessary and not contingent. If something has to exist, then it is part of the very definition of God's nature that he was not created and could not be created.
Consider cosmological arguments for the existence of God for a moment. One type of cosmological argument for God is called a kalam cosmological argument. A kalam cosmological argument follows this format.
1. Whatever begins to exist, must have a cause of its existence.
2. The universe began to exist.
3. Therefore, the universe has a cause of its existence.
In other words, according to the principle used in premise 1, before we know whether we need to ask for a cause of something, we need to discover whether or not, ex hypothesi, it began to exist. If it didn't have a temporal beginning, then a cause may not be needed. This argument is based on the claim that, either through mathematical arguments, or as the upshot of discoveries in astrophysics, we have good reason to suppose that the universe had a temporal beginning (the Big Bang maybe?)
Another cosmological argument for God, found in Aquinas's Third Way, goes like this:
1. Whatever exists contingently must have a cause of its existence.
2. The (physical) universe, and everything in it, exists contingently. It might or might not exist.
Therefore, the universe must have a cause that is independent of the physical universe.
Now, I am not here contending that these are good arguments. However, they are ways of arguing for the existence of God that have been popular amongst philosophers. What I am saying is that these arguments for the existence of God present us with a conception of God that does not need a cause. In fact, if something were to cause God to exist, the God would not be a necessary being, and hence, wouldn't be God.
For this reason, I don't think that the question "Who made God" is the stunning refutation of theism that some people think that it is.