  Exceptional points (EPs) -- spectral singularities of non-Hermitian linear
systems -- have recently attracted great interest for sensing. While initial
proposals and experiments focused on enhanced sensitivities neglecting noise,
subsequent studies revealed issues with EP sensors in noisy environments. Here
we propose a single-mode Kerr-nonlinear resonator for exceptional sensing in
noisy environments. Based on the resonator's dynamic hysteresis, we define a
signal that displays a square-root singularity akin to an EP. In contrast to EP
sensors, our sensor has a signal-to-noise ratio that increases with the
measurement speed, and a precision enhanced at the square-root singularity.
Remarkably, averaging the signal can quickly enhance and then degrade the
precision. These unconventional features open up new opportunities for fast and
precise sensing beyond the constraints of linear systems. While we focus on
optical sensing, our approach can be extended to other hysteretic systems.
