Understanding magnetic field growth in astrophysical objects is a persistent challenge. In stars and galaxies, turbulent flows with net kinetic helicity are believed to be responsible for driving large-scale magnetic fields. However, numerical simulations have demonstrated that such helical dynamos in closed volumes saturate at lower magnetic field strengths when increasing the magnetic Reynolds number Rm. This would imply that helical large-scale dynamos cannot be efficient in astrophysical bodies without the help of helicity outflows such as stellar winds. But do these implications actually apply for very large Rm Here we tackle the long-standing question of how much helical large-scale dynamo growth occurs independent of Rm in a closed volume. We analyze data from numerical simulations with a new method that tracks resistive versus nonresistive drivers of helical field growth. We identify a presaturation regime when the large-scale field grows at a rate independent of Rm, but to an Rm-dependent magnitude. The latter Rm dependence is due to a dominant resistive contribution, but whose fractional contribution to the large-scale magnetic energy decreases with increasing Rm. We argue that the resistive contribution would become negligible at large Rm and an Rm-independent dynamical contribution would dominate if the current helicity spectrum in the inertial range is steeper than k0. As such helicity spectra are plausible, this renews optimism for the relevance of closed dynamos. Our work pinpoints how modest Rm simulations can cause misapprehension of the Rm→∞ behavior.