We present a systematic study of the most luminous ($M_{mathrm{IR}}$ [Vega magnitudes] brighter than $-14$) infrared (IR) transients discovered by the SPitzer InfraRed Intensive Transients Survey (SPIRITS) between 2014 and 2018 in nearby galaxies ($D < 35$ Mpc). The sample consists of nine events that span peak IR luminosities of $M_{[4.5],mathrm{peak}}$ between $-14$ and $-18.2$, show IR colors between $0.2 < ([3.6]{-}[4.5]) < 3.0$, and fade on timescales between $55$ days $< t_{mathrm{fade}} < 480$ 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, multiwavelength follow-up, we suggest the following possible classifications: five obscured core-collapse supernovae (CCSNe), two erupting massive stars, one luminous red nova, and one intermediate-luminosity red transient. 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 eight CCSNe and one Type Iax SN. We find that 7 of the 13 CCSNe in the SPIRITS sample have lower bounds on their extinction of $2 < A_V < 8$. We estimate a nominal fraction of CCSNe in nearby galaxies that are missed by optical surveys as high as $38.5^{+26.0}_{-21.9}$% (90% confidence). This study suggests that a significant fraction of CCSNe may be heavily obscured by dust and therefore undercounted in the census of nearby CCSNe from optical searches.
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