The high-Reynolds-number axisymmetric wake of a slender body with a turbulent boundary layer is investigated using a hybrid simulation. The wake generator is a $6:1$ prolate spheroid and the Reynolds number based on the diameter $D$ is $Re=10^5$. The transition of the wake to a state of complete self-similarity is investigated by looking for the first time into the far field of a slender-body wake. Unlike bluff-body wakes, here the flow is not dominated by vortex shedding in the near wake. Instead, the recirculation region is very small, the near wake is quasi-parallel and is characterised by the presence of broadband turbulence. Until $x/D\approx 20$, the wake decay of a slender body with turbulent boundary layer is very similar to the classic high-$Re$ behaviour, $U_d\sim x^{-2/3}$. Extrapolation of this observation to larger $x/D$ has led to the belief that these wakes decay following the asymptotic $-2/3$ decay law. Our results show, however, that this is not the case and the wake transitions to a faster decay rate once complete self-similarity is achieved. In this later region ($20 < x/D < 80$), mean and turbulence profiles are self-similar. Furthermore, despite the high global and local Reynolds numbers, the classic hypotheses that lead to the well-known decay exponents are not fulfilled. Instead, turbulent dissipation follows a non-equilibrium scaling and a new decay rate $U_d\sim x^{-6/5}$ is observed. The transition from $U_d\sim x^{-2/3}$ to $U_d\sim x^{-6/5}$ is preceded by the dominance of the azimuthal $|m|=1$ mode and the emergence of a large-scale helical structure.