  Our understanding of highly accreting AGNs is hampered by the lack of a
complete systematic investigation in terms of their main spectral and
variability properties, and by the relative paucity of them in the local
Universe, especially those powered by supermassive black holes with
$M_\mathrm{BH} > 10^8\,M_\odot$. To overcome this limitation, we present here
the X-ray spectral analysis of a new, large sample of 61 highly accreting AGNs
named as the \emph{XMM-Newton} High-Eddington Serendipitous AGN Sample, or
X-HESS, obtained by cross-correlating the 11th release of the \emph{XMM-Newton}
serendipitous catalogue and the catalogue of spectral properties of quasars
from the SDSS DR14. The X-HESS AGNs are spread across wide intervals with a
redshift of $0.06<z<3.3$, a black hole mass of
$6.8<\log(M_\mathrm{BH}/M_\odot)<9.8$, a bolometric luminosity of
$44.7<\log(L_\mathrm{bol}/\mathrm{erg\,s}^{-1})<48.3$, and an Eddington ratio
of $-0.2<\log\lambda_\mathrm{Edd}<0.5$, and more than one third of these AGNs
can rely on multiple observations at different epochs, allowing us to
investigate also their variability. We find a large scatter in the $\Gamma -
\lambda_\mathrm{Edd}$ distribution of the highly accreting X-HESS AGNs. A
significant correlation is only found by considering a sample of lower-\ledd\
AGNs with $\lambda_\mathrm{Edd}\lesssim0.3$. The $\Gamma -
\lambda_\mathrm{Edd}$ relation appears to be more statistically sound for AGNs
with lower $M_\mathrm{BH}$ and/or $L_\mathrm{bol}$. We investigate the
possibility of transforming the $\Gamma - \lambda_\mathrm{Edd}$ plane into a
fully epoch-dependent frame by calculating the Eddington ratio from the
simultaneous optical/UV data from the optical monitor,
$\lambda_\mathrm{Edd,O/UV}$. Finally, we also get a mild indication of a
possible anti-correlation between $\Gamma$ and the strength of the soft excess.
