  We derive an approximate analytical expression of the maximum mass of
relativistic self-gravitating Bose-Einstein condensates with repulsive or
attractive $|\varphi|^4$ self-interaction. This expression interpolates between
the general relativistic maximum mass of noninteracting bosons stars, the
general relativistic maximum mass of bosons stars with a repulsive
self-interaction in the Thomas-Fermi limit, and the Newtonian maximum mass of
dilute axion stars with an attractive self-interaction [P.H. Chavanis, Phys.
Rev. D {\bf 84}, 043531 (2011)]. We obtain the general structure of our formula
from simple considerations and determine the numerical coefficients in order to
recover the exact asymptotic expressions of the maximum mass in particular
limits. As a result, our formula should provide a relevant approximation of the
maximum mass of relativistic boson stars for any value (positive and negative)
of the self-interaction parameter. We discuss the evolution of the system above
the maximum mass and consider application of our results to dark matter halos
and inflaton clusters. We also make a short review of boson stars and
Bose-Einstein condensate dark matter halos, and point out analogies with models
of extended elementary particles.
