  We report on the higher-order photon correlations of a high-$\beta$ nanolaser
under pulsed excitation at room temperature. Using a multiplexed four-element
superconducting single photon detector we measured g$^{(n)}(\vec{0})$ with
$n$=2,3,4. All orders of correlation display partially chaotic statistics, even
at four times the threshold excitation power. We show that this departure from
coherence and Poisson statistics is due to the quantum fluctuations associated
with the small number of dipoles and photons involved in the lasing process.
