  How galaxies form, assemble, and cease their star-formation is a central
question within the modern landscape of galaxy evolution studies. These
processes are indelibly imprinted on the galaxy stellar mass function (SMF). We
present constraints on the shape and evolution of the SMF, the quiescent galaxy
fraction, and the cosmic stellar mass density across 90% of the history of the
Universe from $z=7.5\rightarrow0.2$ via the COSMOS survey. Now with deeper and
more homogeneous near-infrared coverage exploited by the COSMOS2020 catalog, we
leverage the large 1.27 deg$^{2}$ effective area to improve sample statistics
and understand cosmic variance particularly for rare, massive galaxies and push
to higher redshifts with greater confidence and mass completeness than previous
studies. We divide the total stellar mass function into star-forming and
quiescent sub-samples through $NUVrJ$ color-color selection. Measurements are
then fitted with Schechter functions to infer the intrinsic SMF, the evolution
of its key parameters, and the cosmic stellar mass density out to $z=7.5$. We
find a smooth, monotonic evolution in the galaxy SMF since $z=7.5$, in
agreement with previous studies. The number density of star-forming systems
seems to have undergone remarkably consistent growth spanning four decades in
stellar mass from $z=7.5\rightarrow2$ whereupon high-mass systems become
predominantly quiescent (i.e. downsizing). An excess of massive systems at
$z\sim2.5-5.5$ with strikingly red colors, some newly identified, increase the
observed number densities to the point where the SMF cannot be reconciled with
a Schechter function. Systematics including cosmic variance and/or AGN
contamination are unlikely to fully explain this excess, and so we speculate
that there may be contributions from dust-obscured objects similar to those
found in FIR surveys. (abridged)
