  The effect of large magnetic Prandtl number $\text{Pm}$ (the ratio of
viscosity to resistivity) on the turbulent transport and energetics of the
magnetorotational instability (MRI) is poorly understood, despite the
realization of this regime in astrophysical environments as disparate as discs
from binary neutron star mergers, the inner regions of low mass X-ray binaries
and active galactic nuclei, and the interiors of protoneutron stars. We
investigate the MRI dynamo and associated turbulence in the regime
$\text{Pm}>1$ by carrying out fully compressible, 3D MHD shearing box
simulations using the finite-volume code \textsc{PLUTO}, focusing mostly on the
case of Keplerian shear relevant to accretion discs. We find that when the
magnetic Reynolds number is kept fixed, the turbulent transport (as
parameterized by $\alpha$, the ratio of stress to thermal pressure) scales with
the magnetic Prandtl number as $\alpha \sim \text{Pm}^{\delta}$, with $\delta
\sim 0.5-0.7$ up to $\text{Pm} \sim 128$. However, this scaling weakens as the
magnetic Reynolds number is increased. Importantly, compared to previous
studies, we find a new effect at very large $\text{Pm}$ -- the turbulent energy
and stress begin to plateau, no longer depending on ${\rm Pm}$. To understand
these results we have carried out a detailed analysis of the turbulent dynamics
in Fourier space, focusing on the effect of increasing $\text{Pm}$ on the
transverse cascade -- a key non-linear process induced by the disc shear flow
that is responsible for the sustenance of MRI turbulence. Finally, we find that
$\alpha$-$\text{Pm}$ scaling is sensitive to the box vertical-to-radial aspect
ratio, as well as to the background shear.
