We present a systematic study of the most luminous (M IR [Vega magnitudes] brighter than −14) infrared (IR) transients discovered by the SPitzer InfraRed Intensive Transients Survey (SPIRITS) between 2014-2018 in nearby galaxies (D < 35 Mpc). The sample consists of 9 events that span peak IR luminosities of M [4.5],peak between −14 and −18.2, show IR colors between 0.2 < ([3.6] − [4.5]) < 3.0, and fade on timescales between 55 < t fade < 480 days. The two reddest events (A V > 12) show multiple, luminous IR outbursts over several years and have directly detected, massive progenitors in archival imaging. With analyses of extensive, multi-wavelength follow-up, we infer the following likely classifications: 5 obscured core-collapse supernovae (CCSNe), 2 erupting massive stars, 1 massive stellar merger, and 1 possible electron capture SN. We define a control sample of all optically discovered transients recovered in SPIRITS galaxies and satisfying the same selection criteria. The control sample consists of 8 core-collapse supernovae (CCSNe) and 1 Type Iax SN. We find that 7 of the 13 CCSNe in the SPIRITS sample have lower bounds on extinction of 2 < A V < 8, suggesting that a large population of CCSNe is heavily obscured. We estimate a nominal fraction of CCSNe in nearby galaxies that are missed by optical surveys of 38.5 +26.0 −21.9 % (90% confidence). This may resolve long-standing discrepancies between measurements of the CCSN rate and the cosmic star-formation rate.