  We investigate two stochastic models of a growing population subject to
selection and mutation. In our models each individual carries a fitness which
determines its mean offspring number. Many of these offspring inherit their
parent's fitness, but some are mutants and obtain a fitness randomly sampled
from a distribution in the domain of attraction of the Fr\'echet distribution.
We give a rigorous proof for the precise rate of superexponential growth of
these stochastic processes and support the argument by a heuristic and
numerical study of the mechanism underlying this growth.
