There is as yet no consensus on the appearance of the large Reynolds number energy spectrum of non-helical MHD turbulence without background field. We use direct numerical simulations with up to 1024^3 meshpoints, together with simulations with hyperdiffusion, in an attempt to see this large Reynolds number limit. We show that the energy spectra consist of a sub-inertial range, around the forcing wavenumber, where the magnetic energy is in sub-equipartition. At larger wavenumbers there is first a range where the magnetic energy is in super-equipartition, followed by a possible equipartition range with a k^(-5/3) slope, before we see a small bottleneck and the diffusive range at the smallest scales. We also investigate the critical magnetic Reynolds number (Rmc) for dynamo action. We find Rmc to be at a minimum for magnetic Prandtl numbers slightly above unity. For magnetic Prandtl numbers smaller than unity we find Rmc=Re^alpha, where alpha is 1/3 for magnetic Prandtl numbers larger than 1/8, and possibly 1 for smaller magnetic Prandtl numbers.