  Observations suggest that massive stellar triples are common. However, their
evolution is not yet fully understood. We investigate the evolution of
hierarchical triples in which the stars of the inner binary experience
chemically homogeneous evolution (CHE), particularly to understand the role of
the tertiary star in the formation of gravitational-wave (GW) sources. We use
the triple-star rapid population synthesis code TRES to determine the evolution
of these systems at two representative metallicities: $Z = 0.005$ and $Z =
0.0005$. About half of all triples harbouring a CHE inner binary (CHE triples)
experience tertiary mass transfer (TMT) episodes, an event which is rare for
classically evolving stars. In the majority of TMT episodes, the inner binary
consists of two main-sequence stars (58-60 per cent) or two black holes (BHs,
24-31 per cent). Additionally, we explore the role of von Zeipel-Lidov-Kozai
(ZLK) oscillations for CHE triples. ZLK oscillations can result in eccentric
stellar mergers or lead to the formation of eccentric compact binaries in
systems with initial outer pericenters smaller than $\sim$ 1200 $R_{\odot}$.
Approximately 24-30 per cent of CHE triples form GW sources, and in 31 per cent
of these, the tertiary star plays a significant role and leads to
configurations that are not predicted for isolated binaries. We conclude that
the evolution of CHE binaries can be affected by a close tertiary companion,
resulting in astronomical transients such as BH-BH binaries that merge via GW
emission orders of magnitude faster than their isolated binary counterparts and
tertiary-driven massive stellar mergers.
