A simple system with infinitely many solutions: a mass orbiting around a fixed gravity potential well. Starting at a certain radius, the mass can orbit the gravitational center with a circle along any direction. They will all reach the opposite end at the same time. Symmetry is probably a general way to have multiple such solutions with specific end points.
A non-symmetry example: a ball bouncing against a wall (a very steep potential wall, without dissipation of course). Given two points in space and time, the ball can go straight to the end with slow speed, or bounce on the potential wall and then go the end point with fast speed. Both trajectories can reach the same end point (same space and time). Symmetry seems to play no role here, as far as I can tell.
In general, The number of solutions depends on specific situations. For instance, two space-like events in special relativity has no solution, as far as I know. Well, no "physical" solution I guess.
From my understand, principle of least action is just a tool to get the equation of motion in classical physics. One doesn't really need to think about varying the path as a minimization problem in most situations. The exceptions might be path integral in quantum mechanics or optics in changing median.