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 4yr 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 the gas envelope of WASP-107b. We reveal that WASP-107b's mass is only 1.8 Neptune masses (Mb=30.5{+/-}1.7M{Earth}). 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 <4.6M{Earth} 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 >~1 au onto a low-opacity, dust-free atmosphere and subsequent migration to the present-day ab=0.0566{+/-}0.0017au. Beyond WASP-107b, we also detect a second, more massive planet (M_c_sini=0.36{+/-}0.04M_J_) on a wide eccentric orbit (ec=0.28{+/-}0.07) that 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.