  We study the full counting statistics (FCS) of quantum gases in samples of
thousands of interacting bosons, detected atom-by-atom after a long free-fall
expansion. In this far-field configuration, the FCS reveals the many-body
coherence from which we characterize iconic states of interacting lattice
bosons, by deducing the normalized correlations $g^{(n)}(0)$ up to the order
$n=6$. In Mott insulators, we find a thermal FCS characterized by
perfectly-contrasted correlations $g^{(n)}(0)= n!$. In interacting Bose
superfluids, we observe small deviations to the Poisson FCS and to the ideal
values $g^{(n)}(0)=1$ expected for a pure condensate. To describe these
deviations, we introduce a heuristic model that includes an incoherent
contribution attributed to the depletion of the condensate. The predictions of
the model agree quantitatively with our measurements over a large range of
interaction strengths, that includes the regime where the condensate is
strongly depleted by interactions. These results suggest that the condensate
component exhibits a full coherence $g^{(n)}(0) =1$ at any order $n$ up to
$n=6$ and at arbitrary interaction strengths. The approach demonstrated here is
readily extendable to characterize a large variety of interacting quantum
states and phase transitions.
