  We present the discovery of TOI-1420b, an exceptionally low-density ($\rho =
0.08\pm0.02$ g cm$^{-3}$) transiting planet in a $P = 6.96$ day orbit around a
late G dwarf star. Using transit observations from TESS, LCOGT, OPM, Whitin,
Wendelstein, OAUV, Ca l'Ou, and KeplerCam along with radial velocity
observations from HARPS-N and NEID, we find that the planet has a radius of
$R_p$ = 11.9 $\pm$ 0.3 $R_\Earth$ and a mass of $M_p$ = 25.1 $\pm$ 3.8
$M_\Earth$. TOI-1420b is the largest-known planet with a mass less than
$50M_\Earth$, indicating that it contains a sizeable envelope of hydrogen and
helium. We determine TOI-1420b's envelope mass fraction to be $f_{env} =
82^{+7}_{-6}\%$, suggesting that runaway gas accretion occurred when its core
was at most $4-5\times$ the mass of the Earth. TOI-1420b is similar to the
planet WASP-107b in mass, radius, density, and orbital period, so a comparison
of these two systems may help reveal the origins of close-in low-density
planets. With an atmospheric scale height of 1950 km, a transmission
spectroscopy metric of 580, and a predicted Rossiter-McLaughlin amplitude of
about $17$ m s$^{-1}$, TOI-1420b is an excellent target for future atmospheric
and dynamical characterization.
