The realization of Bose-Einstein condensation in ultracold trapped gases has led to a revival of interest in this fascinating quantum phenomenon. This experimental achievement necessitated both extremely low temperatures and sufficiently weak interactions. Particularly in reduced spatial dimensionality even an infinitesimal interaction immediately leads to a departure to quasi-condensation. We propose a system of strongly interacting bosons, which overcomes those obstacles by exhibiting a number of intriguing related features: (i) The tuning of just a single control parameter drives a transition from quasi-condensation to complete condensation, (ii) the destructive influence of strong interactions is compensated by the respective increased mobility, (iii) topology plays a crucial role since a crossover from one- to ‘infinite’-dimensionality is simulated, (iv) a ground state gap opens, which makes the condensation robust to thermal noise. Remarkably, all these features can be derived by analytical and exact numerical means despite the non-perturbative character of the system.