  We present a simple and easy-to-use Nash--Moser iteration theorem tailored
for singular perturbation problems admitting a formal asymptotic expansion or
other family of approximate solutions depending on a parameter $\eps\to 0.$ The
novel feature is to allow loss of powers of $\eps$ as well as the usual loss of
derivatives in the solution operator for the associated linearized problem. We
indicate the utility of this theorem by describing sample applications to (i)
systems of quasilinear Schr\"odinger equations, and (ii) existence of
small-amplitude profiles of quasilinear relaxation systems.
