I was musing about this today and couldn't come up with an answer. Obviously it can be generated as a monoid by the four elements $a$, $b$, $a^{-1}$, and $b^{-1}$. After some playing around I was able to come up with three elements that generate it as a monoid: $ab$, $ab^{-1}$, and $a^{-1}$.
But I haven't been able to come up with two generators, nor an argument as to why that should be impossible.