  We calculate the static polarization of AAA-stacked trilayer graphene (TLG)
and study its screening properties within the random phase approximation (RPA)
in all undoped, doped and biased regimes. We find that the static polarization
of undoped AAA-stacked TLG is a combination of the doped and undoped single
layer graphene static polarization. This leads to an enhancement of the
dielectric background constant along a Thomas-Fermi screening with the
Thomas-Fermi wave vector which is independent of carrier concentrations and a
1/r^3 power law decay for the long-distance behavior of the screened coulomb
potential. We show that effects of a bias voltage can be taken into account by
a renormalization of the interlayer hopping energy to a new
bias-voltage-dependent value, indicating screening properties of biased
AAA-stacked TLG can be tuned electrically. We also find that screening
properties of doped AAA-stacked TLG, when $\mu$ exceeds $\sqrt{2}\gamma$, are
similar to that of doped SLG only depending on doping. While for
$\mu<\sqrt{2}\gamma$, its screening properties are a combination of SLG and
AA-stacked screening properties and they are determined by doping and the
interlayer hopping energy.
