Is there any case in physics where the equations of motion depend on high time derivatives of the position?

For example, the Dirac-Lorentz equation.

The radiation-reaction force does not really describe fundamental physics. It's a semi-classical attempt to describe a fundamentally quantum mechanical process. This is why a seemingly simple question: does a uniformly accelerating charge radiate? can lead to almost endless debate. So caveat lector. But it is the standard problem involving jerk, the time derivative of acceleration.