  A massless scalar field exhibits an instability at the event horizon of an
extreme black hole. We study numerically the nonlinear evolution of this
instability for spherically symmetric perturbations of an extreme
Reissner-Nordstrom (RN) black hole. We find that generically the endpoint of
the instability is a non-extreme RN solution. However, there exist fine-tuned
initial perturbations for which the instability never decays. In this case, the
perturbed spacetime describes a time-dependent extreme black hole. Such
solutions settle down to extreme RN outside, but not on, the event horizon. The
event horizon remains smooth but certain observers who cross it at late time
experience large gradients there. Our results indicate that these dynamical
extreme black holes admit a C^1 extension across an inner (Cauchy) horizon.
