The effect of large magnetic Prandtl number 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 measured by the stress-to-thermal-pressure ratio $\alpha$) 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 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 increasing Pm has 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 the scaling of turbulent transport with Pm is sensitive to the box vertical-to-radial aspect ratio, as well as to the background shear: tall boxes exhibit weaker scaling compared to cubic boxes, while MRI turbulence in sub-Keplerian shear flows (characteristic of protoneutron stars) exhibits stronger scaling than it does in Keplerian discs.