Disc galaxies forming in a LambdaCDM cosmology often experience violent mergers. The fact that disc galaxies are ubiquitous suggests that quiescent histories are not necessary. Modern cosmological simulations can now obtain realistic populations of disc galaxies, but it is still unclear how discs manage to survive massive mergers. Here we use a suite of hydrodynamical cosmological simulations to elucidate the fate of discs encountering massive mergers. We follow the changes in the post-merger disc-to-total ratios (D/T) of simulated galaxies and examine the relations between their present-day morphology, assembly history and gas fractions. We find that approximately half of present-day disc galaxies underwent at least one merger with a satellite more massive the host's stellar component and a third had mergers with satellites three times as massive. These mergers lead to a sharp, but often temporary, decrease in the D/T of the hosts, implying that discs are usually disrupted but then quickly re-grow. To do so, high cold gas fractions are required post-merger, as well as a relatively quiescent recent history (over a few Gyrs before z = 0). Our results show that discs can form via diverse merger pathways and that quiescent histories are not the dominant mode of disc formation.