Is there a commutative ring $R$ with two elements $a$ and $b$ for which $b \in \bigcap_{n \ge 0} a^nR$ (the intersection of the principal ideals generated by the powers of $a$), but there is no infinite sequence $(b_n)_{n \ge 0}$ for which $b_0=b$ and $b_n=ab_{n+1}$ for all $n \ge 0$?
Possible example:
Let $R$ be the quotient of the polynomial ring $\mathbb{Z}[x, y_0, y_1, y_2, y_3, ...]$ by the ideal generated by the elements $y_0-x^ny_n$ for $n \ge 1$. Then, $y_0 \in \bigcap_{n \ge 0} x^nR$.
But does $a=x$ and $b=y_0$ work? Showing that $y_n \ne xy_{n+1}$ for $n \ge 1$ (i.e., the obvious sequence does not work) is not enough, as there could be another sequence (other than the obvious one) that may work.