  Slow-roll inflation requires the inflaton field to have an exceptionally flat
potential, which combined with measurements of the scale of inflation demands
some degree of fine-tuning. Alternatively, the flatness of the potential could
be due to the inflaton's origin as a pseudo-Goldstone boson, as in Natural
Inflation. Alas, consistency with Planck data places the original proposal of
Natural Inflation in a tight spot, as it requires a trans-Planckian excursion
of the inflaton. Although one can still tune the renormalizable potential to
sub-Planckian values, higher order corrections from quantum gravity or sources
of breaking of the Goldstone symmetry would ruin the predictivity of the model.
In this paper we show how in more realistic models of Natural Inflation one
could achieve inflation without a trans-Planckian excursion of the field. We
show how a variant of Extra-natural inflation with bulk fermions can achieve
the desired goal and discuss its four-dimensional duals. We also present a new
type of four dimensional models inspired in Little Higgs and Composite Higgs
models which can lead to sub-Planckian values of the inflaton field.
