Consider the sequence $r_n = (1/2)^{1/n!}$. Then $\sum_{k=0}^n r_n^{k!}$ is bounded below by $\sum_{k=0}^n r_n^{n!}$, since $r_n < 1$ and $n! > k!$. Now $r_n^{n!} = 1/2$, so $$\sum_{k=0}^n r_n^{k!} \geq n/2$$
In particular, if we write $S(r)$ to be the sum $\sum_{k=0}^\infty r^{k!}$, this says $S(r_n) \geq n/2$
Therefore $S(r) \to \infty$ as $r \to 1$ - the function $S$ is monotonic and we've exhibited an unbounded increasing subset
