  Numerical simulations of quasi-static magnetoconvection with a vertical
magnetic field are carried out up to a Chandrasekhar number of $Q=10^8$ over a
broad range of Rayleigh numbers $Ra$. Three magnetoconvection regimes are
identified: two of the regimes are magnetically-constrained in the sense that a
leading-order balance exists between the Lorentz and buoyancy forces, whereas
the third regime is characterized by unbalanced dynamics that is similar to
non-magnetic convection. Each regime is distinguished by flow morphology,
momentum and heat equation balances, and heat transport behavior. One of the
magnetically-constrained regimes appears to represent an `ultimate'
magnetoconvection regime in the dual limit of asymptotically-large buoyancy
forcing and magnetic field strength; this regime is characterized by an
interconnected network of anisotropic, spatially-localized fluid columns
aligned with the direction of the imposed magnetic field that remain
quasi-laminar despite having large flow speeds. As for non-magnetic convection,
heat transport is controlled primarily by the thermal boundary layer.
Empirically, the scaling of the heat transport and flow speeds with $Ra$ appear
to be independent of the thermal Prandtl number within the
magnetically-constrained, high-$Q$ regimes.
