In order better to understand the processes that lead to the generation of magnetic fields of finite amplitude, we study dynamo action driven by turbulent Boussinesq convection in a rapidly rotating system. In the limit of infinite Prandtl number (the ratio of viscous to thermal diffusion) the inertia term drops out of the momentum equation, which becomes linear in the velocity. This simplification allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. While the former velocity defines the kinematic dynamo problem responsible for the exponential growth of the magnetic field, the latter encodes the magnetic back reaction that leads to the eventual nonlinear saturation of the dynamo. We argue that two different types of solution should exist: weak solutions in which the saturated velocity remains close to the kinematic one, and strong solutions in which magnetic forces drive the system into a new strongly magnetised state that is radically different from the kinematic one. Indeed, we find both types of solutions numerically. Interestingly, we also find that, in our inertialess system, both types of solutions exist on the same subcritical branch of solutions bifurcating from the non-magnetic convective state, in contrast with the more traditional situation for systems with finite inertia in which weak and strong solutions are thought to exist on different branches. We find that for weak solutions, the force balance is the same as in the non-magnetic case, with the horizontal size of the convection varying as the one-third power of the Ekman number (the ratio of viscous to Coriolis forces), which gives rise to very small cells at small Ekman numbers (i.e. high rotation rates). In the strong solutions, magnetic forces become important and the convection develops on much larger horizontal scales. However, we note that even in the strong cases the solutions never properly satisfy Taylor’s constraint, and that viscous stresses continue to play a role. Finally, we discuss the relevance of our findings to the study of planetary dynamos in rapidly rotating systems such as the Earth.