  We investigate Galton--Watson processes in varying environment, for which
$\bar f_n \uparrow 1$ and $\sum_{n=1}^\infty (1-\bar f_n) = \infty$, where
$\bar f_n$ stands for the offspring mean in generation $n$. Since the process
dies out almost surely, to obtain nontrivial limit we consider two scenarios:
conditioning on non-extinction, or adding immigration. In both cases we show
that the process converges in distribution without normalization to a
nondegenerate compound-Poisson limit law. The proofs rely on the shape function
technique, worked out by Kersting (2020).
