We show that a large class of modified gravity theories (MOG) with the
Jordan-frame Lagrangian $f(R)$ translate into scalar-field (scalaron) models
with hilltop potentials in the Einstein frame. (A rare exception to this rule
is provided by the Starobinsky model for which the corresponding scalaron
potential is plateau-like for $\phi > 0$.) We find that MOG models featuring
two distinct mass scales lead to scalaron potentials that have a flattened
hilltop, or tabletop. Inflationary evolution in tabletop models agrees very
well with CMB observations. Tabletop potentials therefore provide a new and
compelling class of MOG-based inflationary models. By contrast, MOG models with
a single mass scale generally correspond to steep hilltop potentials and fail
to reproduce the CMB power spectrum. Inflationary evolution in hilltop/tabletop
models can proceed in two alternative directions: towards the stable point at
small $R$ describing the observable universe, or towards the asymptotic region
at large $R$. The MOG models which we examine have several new properties
including the fact that gravity can become asymptotically vanishing, with
$G_{\rm eff} \to 0$, at infinite or large finite values of the scalar curvature
$R$. A universe evolving towards the asymptotically vanishing gravity region at
large $R$ will either run into a ‘Big-Rip’ singularity, or inflate eternally.
