  With a mass in the Neptune regime and a radius of Jupiter, WASP-107b presents
a challenge to planet formation theories. Meanwhile, the planet's low surface
gravity and the star's brightness also make it one of the most favorable
targets for atmospheric characterization. Here, we present the results of an
extensive 4-year Keck/HIRES radial-velocity (RV) follow-up program of the
WASP-107 system and provide a detailed study of the physics governing the
accretion of its gas envelope. We reveal that WASP-107b's mass is only 1.8
Neptune masses ($M_b = 30.5 \pm 1.7$ $M_\oplus$). The resulting extraordinarily
low density suggests that WASP-107b has a H/He envelope mass fraction of $>
85$% unless it is substantially inflated. The corresponding core mass of $<4.6$
$M_\oplus$ at 3$\sigma$ is significantly lower than what is traditionally
assumed to be necessary to trigger massive gas envelope accretion. We
demonstrate that this large gas-to-core mass ratio most plausibly results from
the onset of accretion at $\gtrsim 1$ AU onto a low-opacity, dust-free
atmosphere and subsequent migration to the present-day $a_b = 0.0566 \pm
0.0017$ AU. Beyond WASP-107b, we also detect a second more massive planet ($M_c
\sin i = 0.36 \pm 0.04$ $M_{J}$) on a wide eccentric orbit ($e_c = 0.28 \pm
0.07$) which may have influenced the orbital migration and spin-orbit
misalignment of WASP-107b. Overall, our new RV observations and envelope
accretion modeling provide crucial insights into the intriguing nature of
WASP-107b and the system's formation history. Looking ahead, WASP-107b will be
a keystone planet to understand the physics of gas envelope accretion.
