Goedel's incompleteness tells us that any system containing Robinson arithmetic is incomplete. OTOH, Presburger Arithmetic, which contains only the successor and addition, is complete. I'm pretty sure that I have read that it possible to define a complete arithmetic with only sucessor and multiplication axioms. So, this raises the question about what is special about the combination of addition and multiplication.
In reviewing the proof of Goedel's theorem, I thought I had located the key point: the $\beta$-function involves both addition and mutliplication. Without that, we cannot get to the primitive recursive functions, which means we cannot arithmetize syntax, so you cannot form the Goedel sentence, and there is nothing that says it cannot be proved. This isn't a 100% satisfying, but it makes a certain amount of sense.
But then I learned that there are self-verifying theories which can arithmetize syntax despite being much weaker than Robinson arithmetic. So that cannot be right.
Is there, then a simple explanation of why the combination of addition and multiplication is necessary for incompleteness to kick in?