Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.

A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p < 1$ it splits into two copies of itself of size $\varepsilon S$, where $S$ is the original size of the amoeba, and $0 < \varepsilon < 1$ is a fixed number. With probability $q = 1-p$, it doubles in size instead.

Start with one amoeba of size $1$. Denoting by $S^{\max}_t$ the size of the largest amoeba at time $t$, is it true that no matter what the values of $\varepsilon, p$ are, we have

$$\lim_{t \to \infty} S^{\max}_t = \infty$$

almost surely? If not, for what parameter values do we indeed have the desired convergence?