  Aretakis has proved that a massless scalar field has an instability at the
horizon of an extreme Reissner-Nordstr\"om black hole. We show that a similar
instability occurs also for a massive scalar field and for coupled linearized
gravitational and electromagnetic perturbations. We present numerical results
for the late time behaviour of massless and massive scalar fields in the
extreme RN background and show that instabilities are present for initial
perturbations supported outside the horizon, e.g.\ an ingoing wavepacket. For a
massless scalar we show that the numerical results for the late time behaviour
are reproduced by an analytic calculation in the near-horizon geometry. We
relate Aretakis' conserved quantities at the future horizon to the
Newman-Penrose conserved quantities at future null infinity.
