High order derivatives and their impacts to a system

In a simple mass-on-spring system, the motion has time derivatives of all orders. You can compute all of them from $x(t)=Acos{omega t}+Bsin{omega t}$. (Let’s consider the undamped case; the damped one is not conceptually different.) Since $omega$ is expressible in terms of just the spring constant $k$ and the mass $m$, and $A$ and $B$ are determined by the initial position and velocity, all the time derivatives are fully determined. You need to know nothing else about the system.

The equations of classical mechanics do not involve jerk, snap, etc. so you do not generally need to ever think about them. General motion involves nonzero time derivatives of all orders, but that doesn’t change how you find the motion of an object. Newton’s Second Law $F=ma$ is only a second-order differential equation, since $F$ is generally a function of only position and velocity. If you know the force and the mass, you know the acceleration. From only the initial position, initial velocity, and acceleration at each instant, you can find the motion.

If you are doing engineering, you might have constraints on these higher derivatives, but they have essentially nothing to do with physics, until you get to electrodynamics, where the Abraham-Lorentz force depends on the jerk.