  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.
