  In this work we present a complete three-loop renormalisation of an abelian
chiral gauge theory within the Breitenlohner-Maison/'t Hooft-Veltman (BMHV)
scheme of dimensional regularisation (DReg). In this scheme the
$\gamma_5$-matrix appearing in gauge interactions is a non-anticommuting
object, leading to a breaking of gauge and BRST invariance. Employing an
efficient method based on the quantum action principle, we obtain the complete
three-loop counterterm action which serves both to render the theory finite and
to restore gauge and BRST invariance. The UV singular counterterms involve not
only higher order $\epsilon$-poles but also new counterterm structures emerging
at the three-loop level for the first time; the finite symmetry-restoring
counterterms are restricted to the same structures as at lower loop orders,
just with different coefficients, aligning with our expectations. Both the
singular and the finite counterterms include structures which cannot be
obtained by the standard multiplicative renormalisation. Our results
demonstrate that a rigorous treatment of chiral gauge theories with $\gamma_5$
defined in the BMHV scheme at the multi-loop level is possible and that the
obtained counterterm action is suitable for computer implementations, allowing
automated calculations without ambiguities caused by $\gamma_5$.
