  Ultra-luminous quasars ($M_{1450} \leq -29$) provide us with a rare view into
the nature of the most massive and most rapidly accreting supermassive black
holes (SMBHs). Following the discovery of two of these extreme sources,
J0341${+}$1720 ($M_{1450}=-29.56$, $z=3.71$) and J2125${-}$1719
($M_{1450}=-29.39$, $z=3.90$), in the Extremely Luminous Quasar Survey (ELQS)
and its extension to the Pan-STARRS\,1 footprint (PS-ELQS), we herein present
an analysis of their rest-frame UV to optical spectroscopy. Both quasars harbor
very massive SMBHs with
$M_{\rm{BH}}=6.73_{-0.83}^{+0.75}\times10^{9}\,M_{\odot}$ and
$M_{\rm{BH}}=5.45_{-0.55}^{+0.60}\times10^{9}\,M_{\odot}$, respectively,
showing evidence of accretion above the Eddington limit
($L_{\rm{bol}}/L_{\rm{Edd}}=2.74_{-0.27}^{+0.39}$ and
$L_{\rm{bol}}/L_{\rm{Edd}}=3.01_{-0.30}^{+0.34}$). NOEMA 3 millimeter
observations of J0341${+}$1720 reveal a highly star-forming
($\rm{SFR}\approx1500\,M_{\odot}\,\rm{yr}^{-1}$), ultra-luminous infrared
galaxy ($L_{\rm{TIR}}\approx1.0\times10^{13}\,L_{\odot}$) host, which, based on
an estimate of its dynamical mass, is only ${\sim}30$ times more massive than
the SMBH it harbors at its center. As examples of luminous super-Eddington
accretion, these two quasars provide support for theories, which explain the
existence of billion solar mass SMBHs ${\sim}700$ million years after the Big
Bang by moderate super-Eddington growth from standard SMBH seeds.
