As the equations of motion are of second order, the higher derivatives give no new information (but follow uniquely from the initial conditions of position and velocity), therefore they usually are not discussed. (Note: As Timaeus pointed out there are specific scenarios, e.g. Norton's dome where intial values for the higher order derivates will change the outcome, this has to do with singular solutions of ODE, but these are unstable against infinitesimal perturbations, so I would not consider them physical).
The higher derivatives occur in some engineering applicaitons, usually in the context of safety limitations of something. As the jerk determines the rate of change of accelaration it is relevant when some mechanical device must get into an equilibrium with the apparent force due to acceleration. Intuitively, if the jerk is in a way higher than the resonance frequency nasty things may happen (because instead of finding a new equilibrium the system will instead oscillate and may break).
This gets even more important for controlled systems (such as humans) where the regulation may break on large jerk (as it has no sufficient time to reach a new equilibrium state if the setpoint is changing to fast). See Wikipedia on this.
The jerk does occur in an equation of motion in one case in physics: Radiation dampening of motion. See the Lorentz-Abraham force. This seems to cause problems with causality and energy conservation (as the equation has a solution with infinitely increasing energy if the initial jerk has certain properties). In a physical system these problems are avoided because the initial jerk is not actually free, but bound by the previous motion of the system. (This problem comes about because of the infinte energy density of a point charge, so to say directly from the inconsistency of classical electrodynamics – due to special relativity, which follows from classical electrodynamics, particles must be point-like but classical electrodynamics shows divergencies for point charges).