The question is how common or rare are advantageous functional sequences among the possible combinations of the DNA code? The answer is 1x10^77. That’s a 10 with 77 zeros. Those odds are so big that even billions of years could not produce that many outcomes.
I assume you meant to write 1/10^77, in order that we're actually talking about a probability or a frequency.
It's important to consider what this number actually represents. If this is the proportion of combinations of DNA that give rise to functional proteins; then it's not a probability. It would be a probability if we were taking random stretches of DNA and seeing if they coded for functional proteins, but where does that happen?
That's not what it's supposed to be, of course. What Axe was trying to estimate was the likelihood of a protein sequence (not DNA) performing a
specific function. This is an important difference, because you seem to be trying to use it to imply evolution could never happen because useful proteins are too unlikely to form. But even if Axe's calculation is correct (and 1/10^77 is his lower estimate) it only applies to one specific function - is that specific function necessary for life? If not, then why does this matter?
You mentioned another study that claimed 1/10^33 proteins had a useful function. No idea what study that was, but note that this is a difference of 43 orders of magnitude. It's still a very small number, yes, but does it not make you think there might be different things being measured here? If the same thing is being measured, then I'd question whether we know how to measure it.
Here's
another study for you. They created 6,000 million random proteins of 80 amino acids in length. 4 of these proteins bound to ATP. That's 1/10^9 - 68 orders or magnitude different from Axe's estimate. Binding to ATP is, of course, not an essential requirement of a functional protein in any abstract sense - it's just a criterion that makes sense to look for in real organisms.
None of these numbers are much use unless you know what they mean.