I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequance $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\rangle \subsetneqq\langle a_3\rangle \subsetneqq\cdots$ and for each $n\in \mathbb{N}$, $\langle a_i\rangle \subsetneqq\langle a_{i+1}^n\rangle$ for all $i$.