  We study initial conditions for inflation in scenarios where the inflaton
potential has a plateau shape. Such models are those most favored by Planck
data and can be obtained in a large number of model classes. As a
representative example, we consider Higgs inflation with and without an $R^2$
term in the context of Palatini gravity. We show that inflation with a large
number of e-folds generically occurs in a large part of the parameter space
without any fine-tuning of parameters even when the scale of inflation and the
inflaton field value during inflation are much smaller than the Planck scale.
We discuss consequences for detection of primordial gravitational waves and
spectral tilt of curvature perturbations, as well as the recently proposed
"Trans-Planckian Censorship" conjecture.
