  Inspired by the newly observed $T_{cc}^+$ state, we systematically
investigate the $S$-wave triple-charm molecular states composed of $D^*D^*D$
and $D^*D^*D^*$. We employ the one-boson-exchange model to derive the
interactions between $D(D^*)$ and $D^*$ and solve the three-body Schr\"odinger
equations with the Gaussian expansion method. The $S$-$D$ mixing and coupled
channel effects are carefully assessed in our study. Our results show that the
$I(J^P)=\frac{1}{2}(0^-,1^-,2^-)$ $D^*D^*D$ and
$I(J^P)=\frac{1}{2}(0^-,1^-,2^-,3^-)$ $D^*D^*D^*$ systems could form bound
states, which can be viewed as three-body hadronic molecules. We present not
only the binding energies of the three-body bound states, but also the
root-mean-square radii of $D $-$D^*$ and $D^*$-$D^*$, which further corroborate
the molecular nature of these states. These predictions could be tested in the
future at LHC or HL-LHC.
