  By solving exactly the Faddeev equations for the bound-state problem of three
mesons, we demonstrate that current theoretical predictions pointing to the
existence of a deeply-bound doubly bottom axial vector tetraquark lead to the
existence of a unique bound state of three $B$ mesons. We find that the
$BB^*B^*-B^*B^*B^*$ state with quantum numbers $(I)J^P=(1/2)2^-$, $T_{bbb}$, is
about 90\,MeV below any possible three $B$-meson threshold for the reported
binding of the doubly bottom axial vector tetraquark, $T_{bb}$.
