Galaxy merger histories correlate strongly with stellar mass, largely regardless of morphology. Thus, at fixed stellar mass, spheroids and discs share similar assembly histories, both in terms of the frequency of mergers and the distribution of their mass ratios. Since mergers are the principal drivers of disc-to-spheroid morphological transformation, and the most massive galaxies typically have the richest merger histories, it is surprising that discs exist at all at the highest stellar masses (e.g. beyond the knee of the mass function). Using Horizon-AGN, a cosmological hydro-dynamical simulation, we show that extremely massive (M*> 10^11.4 MSun) discs are created via two channels. In the primary channel (accounting for ~70% of these systems and ~8% of massive galaxies) the most recent, significant merger (stellar mass ratio > 1:10) between a massive spheroid and a gas-rich satellite `spins up' the spheroid by creating a new rotational stellar component, leaving a massive disc as the remnant. In the secondary channel (accounting for ~30% of these systems and ~3% of massive galaxies), a system maintains a disc throughout its lifetime, due to an anomalously quiet merger history. Not unexpectedly, the fraction of massive discs is larger at higher redshift, due to the Universe being more gas-rich. The morphological mix of galaxies at the highest stellar masses is, therefore, a strong function of the gas fraction of the Universe. Finally, these massive discs have similar black-hole masses and accretion rates to massive spheroids, providing a natural explanation for why a minority of powerful AGN are surprisingly found in disc galaxies.