Setup

Let

$$ X_0=1, \qquad X_n=\sum_{i=1}^{X_{n-1}} Z_{n,i}, $$

where the offspring variables are i.i.d., nonnegative, and satisfy

$$ \mathbb E[Z]=1. $$

Then $(X_n)$ is a martingale because

$$ \mathbb E[X_n\mid \mathcal F_{n-1}]

X_{n-1}. $$

$X_n\ge 0$, so $(X_n)$ is a nonnegative martingale.
By the martingale convergence theorem, $X_n\to X_\infty$ almost surely.
Since each $X_n$ is integer-valued, convergence implies the sequence is eventually constant.
A positive integer state cannot be absorbing when the offspring law is nondegenerate.

Therefore the only possible limit is

$$ X_\infty=0 \qquad \text{almost surely.} $$

The branching process becomes extinct with probability $1$.