They mostly exist so they can be called "snap", "crackle", and "pop".

https://web.archive.org/web/20180626030437/https://info.aiaa...

The 4th derivative is quite important for good motion control where it is usually called 'snap'. Specifically, it is relevant both for feedforward control design [1] and trajectory planning [2]. As shown in the latter, it is advantageous to design trajectories based on segments of constant snap. Consequently, also including 'snap' in the feedforward signals makes the achieved position profiles notably smoother.

[1] "Control for precision mechatronics" https://doi.org/10.1007/978-3-030-44184-5_100044

[2] "Trajectory planning and feedforward design for electromechanical motion systems" https://doi.org/10.1016/j.conengprac.2004.02.010

Infinitely many exist if the function is infinitely differentiable (aka smooth).

I believe 4th (& possibly higher) is relevant in aeronautics/astronautics and robotics.