  The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic
roughening of stochastically growing interfaces. In one dimension, the KPZ
equation is exactly solvable and its statistical properties are known to an
exquisite degree. Yet recent numerical simulations in the tensionless (or
inviscid) limit of the KPZ equation [Phil. Trans. Roy. Soc. A 380, 20210090
(2022); Phys. Rev. E 106, 024802 (2022)] unveiled a new scaling, with a
critical dynamical exponent $z=1$ different from the KPZ one $z=3/2$. In this
Letter, we show that this scaling is controlled by a fixed point which had been
missed so far and which corresponds to an infinite non-linear coupling. Using
the functional renormalization group (FRG), we demonstrate the existence of
this fixed point and show that it yields $z=1$. We calculate the correlation
function and associated scaling function at this fixed point, providing both a
numerical solution of the FRG equations within a reliable approximation, and an
exact asymptotic form obtained in the limit of large wavenumbers. Both scaling
functions accurately match the one from the numerical simulations.
