What are some experimental verification of the non-linearity of gravitation? According to my (limited) knowledge, all experiments to date probe only situations which can be understood using the linearized version of general relativity. For example, measuring gravitational waves do not require knowledge of non-linear corrections to the metric. 
Friedmann cosmology and the Schwarzschild metric are solutions of the full Einstein's equation. But has there been experimental confirmation of their predictions which differ from predictions due to linearized versions of them?
Any other experimental confirmation of the non-linearity of gravitation?
 A: I would argue that the earliest experimental tests of GR are in fact showing the non-linearity of the theory. Let's focus on the precession of the Mercury.
In the case of the Newton's theory orbits of the planets moving around the Sun shold be Keplerian orbits, which means that they should close. The observations showed that this is not the case, there is a discrepancy of 43 seconds of arc per century.
To solve this problem we use Schwarzschild metric which is the solution to full (non-linearized) Einstein equations.
By using killing fields ($k_1=\partial_t$ and $k2=\partial_{\phi}$) and normalization condition ($u^2=1$) for time-like geodesic we get the equation.
$$ E=\dot{r}^2+\left(1-\frac{2m}{r}\right)\left(1+\frac{L^2}{r^2} \right) $$
If we treat the term with brackets as a potential we can get
$$ V=1-\frac{2m}{r}+\frac{L^2}{r^2}-\frac{2mL^2}{r^3} $$
where first therm is a constant, so doesn't contribute to equations of motion, the second and third therm are just potential of Keplers problem (Newtonian gravitational potential plus centrifugal term).
The most important in this case is the last term which doesn't appear in Newtonian dynamics (which is linearized version of GR) and adds the precession to observed dynamics of Mercury. So from solution of full Einstein equations we get the effect which doesn't originate from linearized version of those equations.
