Why are there only PID controllers, no PIIIDDD (higher order) controllers?
PID is widely used because it's simple and easily tuned. Explanation for each parameter and their influence are somewhat intuitive. They are ubiquitous in industry making it a preferable solution - any designer¹ will preferably chose a solution that factory floor (on field) technicians can tune.

Is it all you can do with classical control? Of course not. For example, if you expect a ramp perturbation in your system, the only way for your control reject it with 0 error on steady state  is to have a double integrator on your controller.

This is part of a more general rule/principle on classical control - for any signal you want to reject, it has to be part of the controller ( 1/s^2 is a ramp signal, which is also a double integrator transfer function)

And with that we are presented with a design choice, the engineering perspective: 
Can you accept the errors of a simpler controller², or do you cope with that difficulty² of forging a working and stable controller that is not easily tuned?

¹ if possible of course
² which may not exist