# Is there any potential energy that is a function of $\dot{x}$?

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: $$V(r,\dot r) = - \frac m 2 ({\omega} \times r)^2 - m\dot r\cdot (\omega\times r)$$