Specifically, I was wondering if the surface was non-compact with infinitely generated free fundamental group, could the surface bundle itself have infinitely generated free fundamental group. In this case, the fundamental group of the bundle is $F_\infty \rtimes_\phi \mathbb{Z}$. On a group theory level, the above can happen; i.e. there exist groups $F_\infty \rtimes_\phi \mathbb{Z} \cong F_\infty$, as was explained to me here:
https://mathoverflow.net/questions/106472/could-f-infty-rtimes-z-be-isomorphic-to-f-infty
However, I'm not sure if this could be carried out with a surface bundle.
Thanks, Kevin