Not if there are no programmers. Users can interact all day long for thousands of years, but in a world devoid of programmers, you won't get more sophisticated programs. Likewise, DNA can be altered randomly for thousands of years and still not produce a more sophisticated gene. Programming requires design.
"DNA is a linear polymer. The sequence is clearly altered randomly in nature in a great variety of ways -- inversions, translocations, base pair substitutions, deletions, etc."
DNA is altered randomly just as copying a program over and over will randomly alter programming code eventually, which is to say that the alterations are always negative and usually destructive of the original designed program.
In no way, shape, or form can you copy an old DOS program enough times for random alterations to turn the offspring programs into Windows NT programs.
...and when all the possible permutations are used up, DNA may no longer replicate. Will have reached our final generation. I suppose that DNA is as finite as anything else.
You don't understand the power of selection. Take a few billion bacteria descended from a single genetically defined progenitor and treat them with a toxin or withold a vital nutrient and you will likely obtain colonies descended from single mutant individuals that can deal with the toxin or synthesize their own previously required nutrient. The genetic change occurred randomly in the large population but the selection pressure (i.e., toxin or lack of nutrient) caused the mutant strains to replace entirely the ancestral type.
You also need to think differently about your computer program example. Computer codes like DOS and Windows are abstract instruction sets and, unlike the genetic code where every sequence of three bases has potential meaning, this will not be true of computer codes. Additionally, nonsense sequences in the genome are effectively ignored or serve a merely structural function. The first time a computer command mutates it comes to a dead stop and has to be fixed by the programmer. The programmer may fix it by restoring the original command or he may replace that command with a new one he took from a different place in that program or from a different program entirely.
The programmer is merely the agent of selection. He is presumably motivated by a desire to make the program more robust or to do more things faster and more easily but he must always keep the program working. He is not an omnipotent designer but must work with what he has learned and may use trial and error approaches extensively at least until the code has evolved to the point where accummulated conventions prohibit innovations to the basic program. At that point modular programming predominates with whole sections of code being adapted for use with existing programs (i.e., add-ins). As the constraints of speed, memory, user interface, and needs change the programs will change too. Not to mention the fact that poorly written codes are abandoned in favor of better ones due to selection in the marketplace. Randomness and selection exist in the process and it matters not at all that humans are making the selections. All that is necessary for you to do is recognize that selection occurs in nature.