  We investigate the upper mass limit predictions of the baryonic mass for
static neutron stars in the context of $f(R)$ gravity. We use the most popular
$f(R)$ gravity model, namely the $R^2$ gravity, and calculate the maximum
baryon mass of static neutron stars adopting several realistic equations of
state and one ideal equation of state, namely that of causal limit. Our
motivation is based on the fact that neutron stars with baryon masses larger
than the maximum mass for static neutron star configurations inevitably
collapse to black holes. Thus with our analysis, we want further to enlighten
the predictions for the maximum baryon masses of static neutron stars in $R^2$
gravity, which, in turn, further strengthens our understanding of the
mysterious mass-gap region. As we show, the baryon masses of most of the
equations of states studied in this paper, lie in the lower limits of the
mass-gap region $M\sim 2.5-5 M_{\odot}$, but intriguingly enough, the highest
value of the maximum baryon masses we found is of the order of $M\sim 3
M_{\odot}$. This upper mass limit also appears as a maximum static neutron star
gravitational mass limit in other contexts. Combining the two results which
refer to baryon and gravitational masses, we point out that the gravitational
mass of static neutron stars cannot be larger than three solar masses, while
based on maximum baryon masses results of the present work, we can
conspicuously state that it is highly likely the lower mass limits of
astrophysical black holes in the range of $M\sim 2.5-3 M_{\odot}$. This, in
turn, implies that maximum neutron star masses in the context of $R^2$ gravity
are likely to be in the lower limits of the range of $M\sim 2.4-3 M_{\odot}$.
Hence our work further supports the General Relativity claim that neutron stars
cannot have gravitational masses larger than $3$$M_{\odot}$.
