It would be big news the principle of least action was experimentally falsified. This, though, is unlikely to happen.
Rather we should expect something similar to the principle of least action to always hold.
Recall, that classical mechanics is deterministic and hence specifying that initial conditions we find a determined trajectory through which the system moves.
This can be alternatively phrased as saying there is a space of possible trajectories that a system can move in and specifying the initial condition means we pick out a specific trajectory.
Now, calculus allows us to pick out specific points on a curve where the gradient is at a minimum, these are the stationary points.
Thus we ought to expect that we can find a functional on the space of trajectories whose stationary point is the desired trajectory. This functional is the Lagrangian.
The physical content is that the Lagrangian is built simply from the difference of the kinetic and potential energy of the system and thus takes a very simple form.
It's probably worth pointing out that the principle is of archaic origin: Hero of Alexandria pointed out in 100AD that light reflecting from a mirror follows the path of least distance, this was rephrased by Damianus in 200AD as the path of least time and then much much later, Fermat generalised this in 1657 to any motion of light follows a path of least time.