  Quantum mechanics is a nonlocal theory, but not as nonlocal as the
no-signalling principle allows. However, there exist quantum correlations that
exhibit maximal nonlocality: they are as nonlocal as any non-signalling
correlations and thus have a local content, quantified by the fraction $p_L$ of
events admitting a local description, equal to zero. Exploiting the link
between the Kochen-Specker and Bell's theorems, we derive, from every
Kochen-Specker proof, Bell inequalities maximally violated by quantum
correlations. We then show that these Bell inequalities lead to experimental
bounds on the local content of quantum correlations which are significantly
better than those based on other constructions. We perform the experimental
demonstration of a Bell test originating from the Peres-Mermin Kochen-Specker
proof, providing an upper bound on the local content $p_L\lesssim 0.22$.
