  We report experimental measurements of the second-order coherence function
$g^{(2)}(\tau)$ of the light emitted by a laser-driven dense ensemble of
$^{87}$Rb atoms. We observe a clear departure from the Siegert relation valid
for Gaussian chaotic light. Measuring intensity and first-order coherence, we
conclude that the violation is not due to the emergence of a coherent field.
This indicates that the light obeys non-Gaussian statistics, stemming from
non-Gaussian correlations in the atomic medium. More specifically, the
steady-state of this driven-dissipative many-body system sustains high-order
correlations in the absence of first-order coherence. These findings call for
new theoretical and experimental explorations to uncover their origin and they
open new perspectives for the realization of non-Gaussian states of light.
