The potential energy one always considers in classical mechanics is $V(\mathbf{x})$, of which (Newtonian) gravity, SHM etc. are clear examples.
Are there examples (in classical mechanics or otherwise) where the potential energy depends explicitly on any time derivative $\dot{x}, \ddot{x} \dots$?
Yes, consider the Lagrangian for a charged particle in an electromagnetic field: $$ L = \frac 12 m v^2 - e(\phi - \vec v\cdot A) $$ where $\vec v$ is the velocity. Also the inertial potential for rotating reference frames: $$ V(r,\dot r) = - \frac m 2 ({\omega} \times r)^2 - m\dot r\cdot (\omega\times r) $$
References
[1] Moreno. G. A, Barrachina. R. O. A velocity-dependent potential of a rigid body in a rotating frame. American Journal of Physics 76, 1146 (2008)
