  General Relativity is recovered from Brans-Dicke gravity in the limit of
large $\omega$. In this article we investigate theories of Brans-Dicke gravity
with chaotic inflation, allowing for either a constant or variable value of
$\omega$, known as extended and hyperextended inflation respectively. The main
focus of the paper is placed on the latter. The variation $\omega$ with respect
to the Brans-Dicke field is based on higher-order corrections analogous to
those of the dilaton field in string theory, following the simple principle
that the Brans-Dicke and metric fields decouple asymptotically. The question
addressed is whether a large value of $\omega$ is predominant in most regions
of the universe, which would lead to the conclusion that a typical region is
then governed by General Relativity. In these theories we find that it is
possible to construct inflaton potentials that drive the evolution of the
Brans-Dicke field to an appropriate range of values at the end of inflation,
such that a large $\omega$ is indeed typical in an average region of the
universe. However, in general this conclusion does not hold and it is shown
that for a wide class of inflaton potentials General Relativity is not a priori
a typical theory.
