  As the era of gravitational-wave astronomy has well and truly begun,
gravitational radiation from rotating neutron stars remains elusive. Rapidly
spinning neutron stars are the main targets for continuous-wave searches since,
according to general relativity, provided they are asymmetrically deformed,
they will emit gravitational waves. It is believed that detecting such
radiation will unlock the answer to why no pulsars have been observed to spin
close to the break-up frequency. We review existing studies on the maximum
mountain that a neutron star crust can support, critique the key assumptions
and identify issues relating to boundary conditions that need to be resolved.
In light of this discussion, we present a new scheme for modelling neutron star
mountains. The crucial ingredient for this scheme is a description of the
fiducial force which takes the star away from sphericity. We consider three
examples: a source potential which is a solution to Laplace's equation, another
solution which does not act in the core of the star and a thermal pressure
perturbation. For all the cases, we find that the largest quadrupoles are
between a factor of a few to two orders of magnitude below previous estimates
of the maximum mountain size.
