The argument from flawed design
[Corrected from an earlier version based on comments. Thanks to my bar-plugta.]
The argument from design begins with the observation that the universe is so intricately complex that only a builder could have created it. The only such builder is the Builder.
An offshoot of this argument is the Intelligent Design theory, which Hirhurim pointed out should better be called a religious philosophy. According to this so-called alternative to evolution, the details of biological systems require a Designer to assemble them all just so.
The problem with both the Intelligent Design philosophy and the argument from design is that complexity need not be explained by a Designer. But the problem of imperfections in the world's design is also relevant. Part of the idea of "complexity" inherent in the Design Argument (and the famous watch metaphor) is that the world's intricacy is necessary to its functioning, in the same way that a complicated mechanism underlies the way a watch works.
An "intelligent design" biologist (to the extent that one exists) would not credit evolution for regulatory proteins and their interlocking complexity. Rather, he would say that this is an instance of special creation, since nothing so complicated as protein-kinase molecular biology could have arisen by chance, even over millions of years.
But the complexity of biological systems is riddled with irregularities and imperfections that weaken the watchmaker metaphor and make us ask the question: what sort of designer would include them? Regulatory proteins can be hijacked by viruses or can malfunction through environmental insult. Even to some nephrologists I know, the kidneys seem like jerry-rigged, hopelessly complicated organs. And if the fine-tuned electrical conduction of the heart goes haywire, you can forget about the ventricles acting as pacemakers on their own. They're likely to conk out altogether, which is deadly to the organism.
Why would God build such imperfections into the system? The test for justifications of God's moral actions, as I think Paul van Inwagen touches upon in one of his lectures, is their narrative plausibility. (The well-known and ancient argument "I believe because it is impossible" can't help us in distinguishing between competing plausible accounts of God's actions.) If we compare the theory of evolution to the theory of special creation of biological systems, the second option presents us with questions that are very difficult to answer: if God specially created molecular biology, why couldn't He (or She) have done a better job? Why doesn't our heart work better? Why are we bipedal when it causes such pain over increased lifespans?
On the other hand, believing in both God and evolution provides a satisfying narrative. God created a system which is higher-order than even humans themselves, a mechanism which governs life in its complexity. This does not gainsay our moral responsibility or particularist relationships with God. Evolution and the Torah are not mutually exclusive. Rather, evolution (so to speak) is a construction more appropriate for the perfect Builder than imperfect special creation.
In other words, the argument from design assumes that complexity requires a Designer. Since not every kind of complexity needs a designer, though, we need to limit the argument to those sorts of complexity which fit the bill. I think that the irregularities and inconsistencies of biological systems -- those which intelligent design advocates hold to be prime examples of special creation -- argue against such "Designing" with a capital D.
A question was raised in the comments: how can we assume that God would design a complexity comprehensible by humans? But that's the very nature of designed complexity! If we point to an irregularity-riddled complexity and say, "This requires a Designer," we are left with very puzzling, perhaps indefensible notions of how the Designer must act.