Does there exist a finite group whose automorphism group is simple?

Since $G/Z(G)$ is the inner automorphism group, and it is normal in $Aut(G)$, this has to be either trivial or all of $Aut(G)$. Thus $G$ is either abelian or a simple complete group. In the abelian case, you've already found a nice set of examples: $Z_2^n$ for $n\ge3$. For the simple complete groups, the Mathieu group $M_{23}$ works.

I can now answer half of my question. If $G = Z_2 \times Z_2 \times Z_2$, then $Aut(G) = GL_3(Z_2) = PSL(3,2)$, which is simple according to this page. I have no insight as to why it is simple. I also have no insight if this is just a freak example, as $|G|=8$ is small in this case. So I would really love a better answer.