"Stripped of all the Turing stuff, this is just the old probabilistic argument, redux. So let me give you my answer. The billions of bases in a human genome did not reach their current state all at once; they evolved from simpler genomes." 
That's an interesting theory, of course, but we simply have no scientific evidence on hand wherein an unguided, unaided, unintelligent "natural" process ever programmed/evolved/created/formed the Base 2 / Base 4 genetic instructions/data for any functional genome (not even a "simple" one).
Furthermore, that sort of theory which you advance above contradicts this: A Tiny Mathematical Proof Against Evolution [AKA - Million Monkeys Can't Type Shakespeare]
So not only does the evidence at hand fail to support your theory above, but so does the math.
That's a pretty weak position in which to place yourself.
That's an interesting theory, of course, but we simply have no scientific evidence on hand wherein an unguided, unaided, unintelligent "natural" process ever programmed/evolved/created/formed the Base 2 / Base 4 genetic instructions/data for any functional genome (not even a "simple" one) Repeating myself: the theory of evolution describes how one organism evolves over time into another. It makes no claims about where the first organism came from. There is a !@#$load of good quantitative evidence of the relationship between the genomes of every living organism that has been studied.
Furthermore, that sort of theory which you advance above contradicts this: A Tiny Mathematical Proof Against Evolution [AKA - Million Monkeys Can't Type Shakespeare]
Probabilistic arguments are almost impossible to apply to complex natural processes. The science of statistical mechanics, which specifically deals with how to calculate probabilities for such processes, can currently handle (at best) problems like how water freezes into ice. We are a hundred years from being able to calculate probabilities of even the simplest possible abiogenetic process, and anyone who says otherwise is simply demonstrating that a little learning is a dangerous thing.
However, if you like probabilities, here's one for you; when salt crystallizes from solution, it goes from a nearly random distribution of sodium ions and chloride ions in water, to a crystal in which there is an exactly ordered array of sodium ions and chloride ions. If you call a sodium 1 and a chloride 0, and you go down one row of the lattice, it looks like this
10101010101010101010
For each atom, there are two choices, so there's a 1 in 2 chance each one will go in correctly.
Now, in a gram of sodium chloride, there are about 2 X 10^22 atoms. So the chances of this happpening corectly in every single case are 0.5 ^ (2 X 10^22). Don't bother with your calculator, it won't go that high, but that's (give or take a few trillion orders of magnitude) a 1 in 10^(5 X 10^21) chance, or 1 in (1 followed by a 5 billion trillion zeroes).
So by the same logic that says evolution can't have happened, salt can't crystallize from salt water. It's just too darn unlikely.
(If you think I'm tring to pull a fast one here, the calculation I just described is a simplified version of the method you would use to calculate the entropy of solution of NaCl from first principles. It's basic chemical physics.)