  We use an effective field theory for short-range forces (SREFT) to analyze
systems of three identical bosons interacting via a two-body potential that
generates a scattering length, $a$, which is large compared to the range of the
interaction, $\ell$. The amplitude for the scattering of one boson off a bound
state of the other two is computed to next-to-next-to-leading order (N$^2$LO)
in the $\ell/a$ expansion. At this order, two pieces of three-body data are
required as input in order to renormalize the amplitude (for fixed $a$). We
apply our results to a model system of three Helium-4 atoms, which are assumed
to interact via the TTY potential. We generate N$^2$LO predictions for
atom-dimer scattering below the dimer breakup threshold using the bound-state
energy of the shallow Helium-4 trimer and the atom-dimer scattering length as
our two pieces of three-body input. Based on the convergence pattern of the
SREFT expansion, as well as differences in the predictions of two
renormalization schemes, we conclude that our N$^2$LO phase- shift predictions
will receive higher-order corrections of $< 0.2$%. In contrast, the prediction
of SREFT for the binding energy of the "deep" trimer of Helium-4 atoms displays
poor convergence.
