  We propose a new approach to find magnetically-dominated force-free
magnetospheres around highly spinning black holes, relevant for models of
astrophysical jets. Employing the near-horizon extreme Kerr (NHEK) limit of the
Kerr black hole, any stationary, axisymmetric and regular force-free
magnetosphere reduces to the same attractor solution in the NHEK limit with
null electromagnetic field strength. We use this attractor solution as the
universal starting point for perturbing away from the NHEK region in the
extreme Kerr spacetime. We demonstrate that by going to second order in
perturbation theory, it is possible to find magnetically dominated
magnetospheres around the extreme Kerr black hole. Furthermore, we consider the
near-horizon near-extreme Kerr (near-NHEK) limit that provides access to a
different regime of highly spinning black holes. Also in this case we find a
novel force-free attractor, which can be used as the universal starting point
for a perturbative construction of force-free magnetospheres. Finally, we
discuss the relation between the NHEK and near-NHEK attractors.
