As far as I can tell, a trajectory with zero excess velocity would place it (essentially) in nearly the same orbit around the sun as earth (assuming it even departs far away enough for the sun’s gravity well to dominate). I can’t think of any possible mission objectives that would make such a trajectory desirable, but perhaps my imagination is limited. Why would one want to choose zero excess velocity upon escape?
Has anyone ever attempted/achieved such a trajectory? If so, what were the mission objectives that made such an escape desirable over a traditional hyperbolic escape?