The exponential map associated with the (-) - connection on a Lie group is generally not surjective. This is because, for this connection, the one-parameter subgroups and geodesics coincide. If we drop the requirement that one-parameter subgroups are geodesics, is it possible to put a left invariant connection on a Lie group for which the geodesic exponential map is surjective?
Yes. Just take the Levi-Civita connection of any left-invariant Riemannian metric on the Lie group. The metric is complete, so any two points can be joined by a geodesic (Hopf-Rinow). Thus, the geodesic exponential map of that connection starting from the identity is surjective.