  The construction of a generic parametrization of spinning geometries which
can be matched continuously to the Kerr metric is an important open problem in
General Relativity. Its resolution is more than of academic interest, as it
allows to parametrize and quantify possible deviations from the no-hair
theorem. Various approaches to the problem have been proposed, all with their
own (severe) limitations. Here we discuss the metric recently proposed by
Johannsen and Psaltis, showing that (i) the original metric describes only
corrections that preserve the horizon area-mass relation of nonspinning
geometries; (ii) this unnecessary restriction can be relaxed by introducing a
new parameter that in fact dominates in both the post-Newtonian and
strong-field regimes; (iii) within this framework, we construct the most
generic spinning black-hole geometry which contains twice as many the
(infinite) parameters of the original metric; (iv) in the strong-field regime,
all parameters are roughly equally important. This fact introduces a severe
degeneracy problem in the case of highly-spinning black holes. Finally, we
prove that even our generalization fails to describe the few known spinning
black-hole metrics in modified gravity.
