Asymptotics of iterated branching processes

We study the iterated Galton-Watson process (IGW), possibly with thinning, introduced by Gawe{\l}and Kimmel to model the number of repeats of DNA triplets during some genetic disorders. If the process involves some thinning, then extinction and explosion can have positive probability simultaneously. If the underlying (simple) Galton-Watson process is nondecreasing with mean m, then, conditionally on the explosion, the logarithm of the population of the IGW at time n+1 is equivalent to log(m) times the population at time n, almost surely. This simplifies arguments of Gawe{\l}and Kimmel, and confirms and extends a conjecture of Pakes.