The non-Hermitian skin effect is an intriguing physical phenomenon, in which all eigenmodes of a non-Hermitian lattice become localized at boundary regions. While such an exotic behavior has been demonstrated in various physical platforms, most realizations have been so far restricted to the linear regime. Here, we explore the cooperation between nonlinearity and the non-Hermitian skin effect, revealing extraordinary amplitude-driven skin localization dynamics. By introducing an extension to the Hatano-Nelson model where couplings inherit nonlinear behavior, we demonstrate the existence of unique amplitude-driven non-Hermitian skin modes capable of concentrating the energy of a source at any point of space, depending on the power level. Our theoretical model is supported by numerical simulations and experimental realization via a highly configurable acoustic metamaterial composed of active electroacoustic resonators. In all, our findings open up exciting paths for a new generation of nonreciprocal systems, in which nonlinearities serve as a strategic tuning knob to manipulate the guiding process.