Fix $p>0$ and define a recursive sequence of random variables with $X_1 =1$ and
$$X_{k+1} = X_k + \text{Bin}(X_k,p).$$
Thus, $\mathbf E [ X_k ] = (1+p)^k$. 
I would like a left tail bound. Perhaps, for some $a>0$ and $0 < \epsilon < p$,
$$\mathbf P[X_k < (1+ p - \epsilon)^k ] \leq e^{-a k}.$$