  Gravitational-wave tails are due to the backscattering of linear waves onto
the space-time curvature generated by the total mass of the matter source. The
dominant tails correspond to quadratic non-linear interactions and arise at the
one-and-a-half post-Newtonian (1.5PN) order in the gravitational waveform. The
"tails-of-tails", which are cubic non-linear effects appearing at the 3PN order
in the waveform, are also known. We derive here higher non-linear tail effects,
namely those associated with quartic non-linear interactions or
"tails-of-tails-of-tails", which are shown to arise at the 4.5PN order. As an
application, we obtain at that order the complete coefficient in the total
gravitational-wave energy flux of compact binary systems moving on circular
orbits. Our result perfectly agrees with black-hole perturbation calculations
in the limit of extreme mass ratio of the two compact objects.
