  Conditions for almost sure extinction are studied in discrete time branching
processes with an infinite number of types. It is not assumed that the expected
number of children is a bounded function of the parent's type. There might also
be no integer m such that there is a lower positive bound, uniform over the
ancestor's type, for the probability that a population is extinct at the m-th
generation. A weaker condition than the existence of such an m is seen to lead
to extinction almost surely if the sequence of expected generation sizes does
not tend to infinity. Some criteria for a positive probability of nonextinction
are given. Examples are provided by extending to our setting two applications,
namely Leslie population dynamics and processes arising in continuum
percolation in which the offsprings follow Poisson point distributions.
