Inspired by comments to answer for this question:
Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$
with initial conditions $x(0)=1$, $\dot x(0)=0$.
If $g(v)=-cv$, then critical damping is when $c=2$. For such $c$, $x(t)$ approaches $0$ at the fastest rate.
Now let $g(v)$ be an arbitrary function. I've tried speeding up approach to $0$ by adding some odd powers of $v$ to it, empirically (and numerically solving the equation) finding that adding all odd powers starting with $3$ with the same constant of $2$ makes approach the fastest. I.e. in this case my function looked like (after summing the series): $$g(v)=-2v+2\frac{v^3}{1-v^2}.$$
Here's how $x(t)$ looks in the case of critical damping (blue) and my best found nonlinear damping (purple):
Is this indeed the best possible damping function? If so, how to prove this? If not, what is the best one (and does it exist — maybe the approaching speed is generally unbounded)?