The majority of massive stars are expected to exchange mass or merge with a companion during their lives. This immediately implies that most supernovae (SNe) are from such post-mass-exchange objects. Here, we explore how mass accretion and merging affect the pre-SN structures of stars and their final fates. To this end, we modelled these complex processes by rapid mass accretion onto stars of different evolutionary stages and followed their evolution up to iron core collapse. We used the stellar evolution code MESA and inferred the outcome of core-collapse using a neutrino-driven SN model. Our models cover initial masses from 11 to 70 M⊙ and the accreted mass ranges from 10−200% of the initial mass. All models are non-rotating and for solar metallicity. The rapid accretion model offers a systematic way to approach the landscape of mass accretion and stellar mergers. It is naturally limited in scope and serves as a clean zeroth order baseline for these processes. We find that mass accretion, in particular onto post-main-sequence (post-MS) stars, can lead to a long-lived blue supergiant (BSG) phase during which stars burn helium in their cores. In comparison to genuine single stars, post-MS accretors have small core-to-total mass ratios, regardless of whether they end their lives as BSGs or cool supergiants (CSGs), and they can have genuinely different pre-SN core structures. As in single and binary-stripped stars, we find black-hole (BH) formation for the same characteristic CO core masses MCO of ≈7 M⊙ and ≳13 M⊙. In models with the largest mass accretion, the BH formation landscape as a function of MCO is shifted by about 0.5 M⊙ to lower masses, that is, such accretors are more difficult to explode. We find a tight relation between our neutron-star (NS) masses and the central entropy of the pre-SN models in all accretors and single stars, suggesting a universal relation that is independent of the evolutionary history of stars. Post-MS accretors explode both as BSGs and CSGs, and we show how to understand their pre-SN locations in the Hertzsprung-Russell (HR) diagram. Accretors exploding as CSGs can have much higher envelope masses than single stars. Some BSGs that avoid the luminous-blue-variable (LBV) regime in the HR diagram are predicted to collapse into BHs of up to 50 M⊙, while others explode in SNe and eject up to 40 M⊙, greatly exceeding ejecta masses from single stars. Both the BH and SN ejecta masses increase to about 80 M⊙ in our models when allowing for multiple mergers, for example, in initial triple-star systems, and they can be even higher at lower metallicities. Such high BH masses may fall into the pair-instability-SN mass gap and could help explain binary BH mergers involving very massive BHs as observed in GW190521. We further find that some of the BSG models explode as LBVs, which may lead to interacting SNe and possibly even superluminous SNe.