We consider the effect of backreaction of quantized massive fields on the
metric of extreme black holes (EBH). We find the analytical approximate
expression for the stress-energy tensor for a scalar (with an arbitrary
coupling), spinor and vector fields near an event horizon. We show that,
independent of a concrete type of EBH, the energy measured by a freely falling
observer is finite on the horizon, so that quantum backreaction is consistent
with the existence of EBH. For the Reissner-Nordstrom EBH with a total mass
M_{tot} and charge Q we show that for all cases of physical interest M_{tot}<
Q. We also discuss different types of quantum-corrected Bertotti-Robinson
spacetimes, find for them exact self-consistent solutions and consider
situations in which tiny quantum corrections lead to the qualitative change of
the classical geometry and topology. In all cases one should start not from a
classical background with further adding quantum corrections but from the
quantum-corrected self-consistent geometries from the very beginning.Comment: Minor corrections. To appear in Phys. Rev.