I want to show a scholar example of a system such that its linearization is not controllable but the system can be stabilized with nonlinear feedback. I am thinking about this one $$ \begin{aligned} \dot{x}_1 &= x_2^3, \\ \dot{x}_2 &= u, \end{aligned} $$ where $x_1$, $x_2$, and $u$ are scalars, and the goal is to drive the system to the origin.
What is the simplest control that you would propose for this system?
Even though the linearization is not controllable, the nonlinear system can still be stabilized with linear feedback. I propose the control law
$$ u = -x_1 - x_2 \tag{1} $$
which leads to the closed loop dynamics
$$ \begin{align} \dot{x}_1 &= x_2^3 \\ \dot{x}_2 &= -x_1 - x_2 \end{align} \tag{2} $$
Take the Lyapunov function
$$ V(x) = x_1^2 + 2 x_1 x_2 + x_2^2 + \frac{1}{2} x_2^4 $$
It is easy to show that $V$ has a unique minimum at $(0, 0)$ so it is positive definite. The derivative is
$$ \dot{V}(x) = -2 (x_1 + x_2)^2 $$
which is negative semi-definite (zero along the $x_1 = -x_2$ line). If we insert that into $(2)$, we have $\dot{x}_2 = 0$ but $\dot{x}_1 = -x_1^3$, so no solution can stay in the set $\dot{V}(x) = 0$ except $x_1 = x_2 = 0$.
So, by LaSalle, the system is globally asymptotically stabilized by the linear feedback $(1)$.
This is probably also the "simplest" stabilizing control law (linear feedback with both gains being 1), but that depends on your definition of simple.