Let $R$ be an integral domain. $I\not= \{ 0 \} $ a maximal ideal.
Is there an example where $I \supseteq I^2 \supseteq \ldots $ satisfies $I^n=I^{n+1} = \cdots $ for all $n \in \mathbb{N}$.
I can show for Noetherian $R$ this cannot be the case. I am unsure in general.
For an example other than the zero ideal of a field, consider the field of rational functions $F(x_1, x_2, x_3,\ldots)$ and take the subring $R$ generated by elements of the form $\dfrac{x_{i+1}}{x_i}$ for $i\in\mathbb N$ and $x_1$.
$R$ is an integral domain, and the ideal $M$ generated by the elements above contains $x_1, x_2, \ldots$ for every $x_i$, so the quotient by that ideal is isomorphic to $F$. That maximal ideal has the property that $M^2=M$.