It seems to me that, as soon as there is mass in the universe which curves space-time, all paths of movement are closed. To a first approximation, for anything close to a single large mass that means an elliptical path.

That assumes, of course, that an instantaneous "path of movement" is something that is even worth considering, given that it ignores the time dimension and the fact that all those other objects aren't remaining in a static configuration around the object whose path we are considering.