  We study an iterated temporal and contemporaneous aggregation of $N$
independent copies of a strongly stationary subcritical Galton-Watson branching
process with regularly varying immigration having index $\alpha \in (0, 2)$.
Limits of finite dimensional distributions of appropriately centered and scaled
aggregated partial sum processes are shown to exist when first taking the limit
as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is
an $\alpha$-stable process if $\alpha \in (0, 1) \cup (1, 2)$, and a
deterministic line with slope $1$ if $\alpha = 1$.
