For more than fourty years, the quest to understand how large-scale magnetic fields emerge from turbulent flows in rotating astrophysical systems, such as the Sun, has been a major thread of computational astrophysics research. Using a parameter scan and phenomenological analysis of maximally-simplified three-dimensional cartesian magnetohydrodynamic simulations of large-scale nonlinear helical turbulent dynamos, I present results in this Letter that strongly point to an asymptotic ultimate regime of the large-scale solar dynamo, at large magnetic Reynolds numbers $Rm$, involving helicity fluxes between hemispheres. I obtained corresponding numerical solutions at both $Pm>1$ and $Pm<1$, and show that they can currently only be achieved in clean, simplified numerical setups. The analysis further strongly suggests that all global simulations to date lie in a non-asymptotic turbulent MHD regimes highly sensitive to changes in kinetic and magnetic Reynolds numbers. Ideas are presented to attempt to reach this ultimate regime in such "realistic" global spherical models at a reasonable numerical cost. Overall, the results clarify the current state, and some hard limitations of the brute-force numerical modelling approach applied to this, and other similar astrophysical turbulence problems.
