x86 program to do an exponent
I'm no expert, but what the heck, here's a review comment:
# needs to be 16-byte aligned before the next function call so do two pushes
pushq $0
pushq %rax

This comment is good, but I would rephrase it. Two eight-byte pushes makes 16 bytes and doesn't change the stack alignment. Therefore I infer that one of these pushes is significant and the other one is insignificant — but your comment doesn't tell me which is which! So you might say instead
# one extra push to preserve 16-byte stack alignment
pushq $0

# push the result of `exp`
pushq %rax

You could make the generated code smaller by eliminating the insignificant constant $0:
# push the result of `exp`, plus one extra push to preserve 16-byte stack alignment
pushq %rax
pushq %rax

Now the reader doesn't even need to figure out which push is the significant one, because both pushes do the same thing!

But why is preserving 16-byte alignment on calls important? That's not a requirement of the machine. You seem to be trying to follow some specific ABI, like maybe for interoperability with C or C++. Your external documentation should be clearer about what ABI you're trying to follow.
And then, if you are trying to interoperate with C code, you could improve your code by indicating which of its labels are meant as external entrypoints and which ones are just internal local labels. It seems like you intend exp to be called from other code — it's an entrypoint — but e.g. exp_op is not callable, and exp_ret is technically callable but just acts as a no-op. You might mark them somehow as &quot;local implementation details, not for external consumption.&quot;
Yeah, you technically already do this by exporting .globl _start and not .globl exp — but there's still a big difference between the callable function exp and the local label exp_op which is not reflected in your naming scheme. If I were doing this, I'd add .globl exp and I'd rename exp_op, exp_ret to something like Lexp1, Lexp2 or L1_looptop, L2_loopend.