This particular item deals with excluding strings that have a consecutive repeated sequence. That is, ab*b*a and abcab*ab*c both have doubles whereas abcabacba doesn't. When working in base $2$, the number of sequences is easily seen to be finite.
They also mention that base $10$ is known to be infinite. In addition, base $3$ seems to be infinite, as evidenced by this exponentially-growing pattern.
My question is: how can you prove that the number of repetition-free sequences is infinite for base $b \ge 3$?