  In this article, we study the mass spectrum of the low-lying triply heavy
hybrid baryon, which consists of three valence heavy quarks in a color octet
and one valence gluon, with spin-parity $J^P=(\frac{1}{2})^+$ via QCD sum
rules. This is the first study on the triply heavy hybrid baryons in the
framework of QCD sum rules. After performing the QCD sum rule analysis, we find
that the mass of $cccg$ hybrid baryon lies in $M_{cccg}= 5.91-6.13$ GeV. As a
byproduct, the mass of the triply bottom hybrid baryon state is extracted to be
around $M_{bbbg}=14.62-14.82$ GeV. The contributions up to dimension eight at
the leading order of $\alpha_s$ (LO) in the operator product expansion are
taken into account in the calculation. The triply charmed hybrid baryon
predicted in this work can decay into one doubly charmed baryon and one charmed
meson. Especially, we propose to search for $cccg$ hybrid baryon with $J^{P}=
(1/2)^+$ in the P-wave decay channels $\Xi_{cc}^{++} D^0$, $\Xi_{cc}^{+} D^+$,
and $\Xi_{ccs}^{+} D_s^+$, which may be accessible in future BelleII, Super-B,
PANDA, and LHCb experiments.
