  Highly degenerate chaotic radiation has a Gaussian density matrix and a large
occupation number of modes $f $. If it is passed through a weakly transmitting
barrier, its counting statistics is close to Poissonian. We show that a second
identical barrier, in series with the first, drastically modifies the
statistics. The variance of the photocount is increased above the mean by a
factor $f$ times a numerical coefficient. The photocount distribution reaches a
limiting form with a Gaussian body and highly asymmetric tails. These are
general consequences of the combination of weak transmission and multiple
scattering.
