The question "who designed the designer" is either a really good question, or a really awful one. For someone just starting to think about these matters, it is an obvious place to begin. For a scientist, such as Richard Dawkins, who asks it as if merely posing the question presents an insurmountable obstacle to ID, it is very height of arrogance if not stupidity.
Let's begin with IDers who claim the designer doesn't have to be God (a point of view that, personally, I don’t find tenable.) If they use Behe's irreducible complexity as the basis for their ID (a theory about which I am ambivalent) then the answer is simple: our immediate designer (if it is not God) was either designed by another creature (or by God) or, ironically, it evolved. That is, while life on earth displays irreducible complexity, there is no need to assume that the creature-designer does. Maybe the creature-designer, examined microscopically and historically, displays clear evidence of having evolved, with no issues of irreducible complexity, or insufficient time, or convergence, or any of the other problems, apparent or not, faced by evolution on earth.
On the other hand, if the designer is God, then the question is meaningless. Only things that are created have to have a cause. The universe must have a cause. Life must have a cause. But something not created does not have a cause. So the question "who designed God?" is ill-formed. Similar to the trick question "have you stopped beating your wife?" it assumes facts not in evidence.
Now, those on the anti-ID side will immediately say "that's a copout!" However, I'll remind them that, when it suited science, science gave the same answer. When the standard model of the universe was the Steady State model, it was perfectly acceptable to respond to the question, "what caused the universe?" with the answer "nothing caused the universe because it wasn't created, it just is."
So when applied to the universe, the answer "it just is" was acceptable, but when applied to God, it apparently isn't. So we have hyper-bigots like Dawkins (who believes Christian parents are child-abusers, if they rear their children in the faith1) asking the question "who made God?" 2 as if it were a show-stopper when in fact it is a trivial question.
Some cosmologists, by the way, theorize that although our universe has a beginning, the universe creation machine that bubbles up universes from the vacuum is eternal. Anyone who asks "who designed the designer" should just as readily, and with the same level of disdain, ask "who or what made the vacuum eternally pregnant with new universes?"
1 "Odious as the physical abuse of children by priests undoubtedly is, I suspect that it may do them less lasting damage than the mental abuse of having been brought up Catholic in the first place.", Richard Dawkins, "Religion's Real Child Abuse", Free Inquiry, Fall 2002, Vol. 22, No. 4., p. 9
2 Richard Dawkins, The Blind Watchmaker : Why the Evidence of Evolution Reveals a Universe without Design, Norton, W. W. & Company, 1986.