The problem is that there's different types of multiplication. The fact that multiplication over the Reals can be thought of as "iterated addition" is a consequence of commutativity. But mathematicians often generalize multiplication to various contexts as an arbitrarily-defined transformation, which may or may not be commutative. E.g. multiplication over the Reals is commutative; multiplication over Complex Numbers is commutative; multiplication over Quaternions is non-commutative.

Incidentally, "exponentiation is just repeated multiplication", right? Try using iteration to solve 0^0.

  lim{x -> 0} (0^x) = 0
  lim{x -> 0} (x^0) = 1

The "iteration mindset" doesn't always generalize cleanly.