  We study the statistics of scalar perturbations in models of inflation with
small and rapid oscillations in the inflaton potential (resonant
non-Gaussianity). We do so by deriving the wavefunction
$\Psi[\zeta(\boldsymbol{x})]$ non-perturbatively in $\zeta$, but at first order
in the amplitude of the oscillations. The expression of the wavefunction of the
universe (WFU) is explicit and does not require solving partial differential
equations. One finds qualitative deviations from perturbation theory for $
|\zeta| \gtrsim \alpha^{-2}$, where $\alpha \gg 1$ is the number of
oscillations per Hubble time. Notably, the WFU exhibits distinct behaviours for
negative and positive values of $\zeta$ (troughs and peaks respectively). While
corrections for $\zeta <0$ remain relatively small, of the order of the
oscillation amplitude, positive $\zeta$ yields substantial effects, growing
exponentially as $e^{\pi\alpha/2}$ in the limit of large $\zeta$. This
indicates that even minute oscillations give large effects on the tail of the
distribution.
