  The average result of a weak measurement of some observable $A$ can, under
post-selection of the measured quantum system, exceed the largest eigenvalue of
$A$. The nature of weak measurements, as well as the presence of post-selection
and hence possible contribution of measurement-disturbance, has led to a
long-running debate about whether or not this is surprising. Here, it is shown
that such "anomalous weak values" are non-classical in a precise sense: a
sufficiently weak measurement of one constitutes a proof of contextuality. This
clarifies, for example, which features must be present (and in an experiment,
verified) to demonstrate an effect with no satisfying classical explanation.
