  The emergence of the Efimov effect in the $D^*D^*D^*$ system is explored
under the assumption that the heavy partner of the $T_{cc}^+$ exists as a
$D^*D^*$ molecule with $(I)J^P=(0)1^+$. The three-to-three relativistic
scattering amplitude is obtained from the ladder amplitude formalism, built
from an energy-dependent contact two-body potential where the molecular
component of the $T_{cc}^*$ state can be varied. We find that
$(I)J^P=(\tfrac{1}{2})0^-$ three-body bound states can be formed, with
properties that suggest that the Efimov effect can be realised for reasonable
values of the molecular probability and binding energy of the $T_{cc}^*$.
