Is there a consistent, complete axiom system that proves its own consistency?
I know that this question isn't exact and I haven't defined when an axiom system proves its own consistency because that's just human interpretation.
Dan Willard published several papers about this topic in the Journal of Symbolic Logic. One place to start is the short Wikipedia article "Self-verifying theories". I am not familiar with the detailed proofs about Willard's theories, but when I have heard him talk about them he indicated they do not prove that multiplication is a total function, and in that way manage to remain weak enough to avoid the incompleteness theorem.