I was surprised when, in The God Delusion, Richard Dawkins dismisses the Cosmological Argument by asking the title question: "Who made God?" Here's how Peter Kreeft briefly responds to the question:
The question "If God made everything, who made God?" is like asking "Who made circles square?" It assumes a self-contradiction: that an uncreated Creator is a created creature. It extends the law about changing things -that every change needs a cause- beyond its limits, to the unchanging Source of change. God does not need a cause, or a maker, because he is not made or changed. He changes other things, but is not himself changed by anything. There is nothing that comes to be in him, nothing that needs a cause for its coming-into-being. (Peter Kreeft, Handbook of Christian Apologetics, 105)
So essentially, Kreeft suggests that the question commits a category error; it overextends the general law that "things that exist require a maker" from the physical, created world, to the non-physical uncreated God.
A further question that may arise: "If God can be 'uncreated & unchanging', why not the universe too?" The answer would be that we have good reasons to believe that the universe is not eternal, as per the Cosmological Argument (including at least scientific and philosophical reasons), while no such restrictions would apply to God.