  Motivated by a seminal paper of Kesten et al. (1975) we consider a branching
process with a geometric offspring distribution with i.i.d. random
environmental parameters $A_n$, $n\ge 1$ and size -1 immigration in each
generation. In contrast to above mentioned paper we assume that the environment
is long-tailed, that is that the distribution $F$ of $\xi_n := \log
((1-A_n)/A_n)$ is long-tailed. We prove that although the offspring
distribution is light-tailed, the environment itself can produce extremely
heavy tails of the distribution of the population size in the n-th generation
which becomes even heavier with increase of n. More precisely, we prove that,
for any n, the distribution tail $\mathbb{P}(Z_n > m)$ of the $n$-th population
size $Z_n$ is asymptotically equivalent to $n\overline{F}(\log m)$ as $m$
grows. In this way we generalize Bhattacharya and Palmowski (2019) who proved
this result in the case $n=1$ for regularly varying environment $F$ with
parameter $\alpha >1$. Further, for a subcritical branching process with
subexponentially distributed $\xi_n$, we provide the asymptotics for the
distribution tail $\mathbb{P}(Z_n>m)$ which are valid uniformly for all $n$,
and also for the stationary tail distribution. Then we establish the "principle
of a single atypical environment" which says that the main cause for the number
of particles to be large is a presence of a single very small environmental
parameter $A_k$.
