Do physicists ever have to account for the rate of change of acceleration? What about the rate of change of the rate of change of acceleration? This could keep going on forever, but is there some point where it just doesn't matter for a fairly accurate estimate?
It's called "jerk", and it's indirectly important in the design of the suspension system and drivetrains of vehicles, and any environment where the rate of change of acceleration corresponds to the rate of change of force or torque. The reaction to any force by an object is some type of deformation, and since it takes time for the deformation reaction to occur, there's a period of time where the object experiences internal linear (suspension) and/or angular (drivetrain) accelerations and deformations. Reducing the "jerk" reduces the shock to these components. For most situations, I think it stops at "jerk".
If there were any higher rates then then the base quantity T(time) would be raised to the powers of 4,5 etc and apart from black body radiation I can't, at present, think of any equations where a base quantity is raised to a power higher than three.It's an interesting question.