Let $R = k[x_0,\dots]$ a polynomial ring over a field with infinitely many variables, and put $\mathfrak{m} = (x_0,\dots)$ the ideal generated by degree one homogeneous polynomials(that is, the variables). Put $A = R_\mathfrak{m}$ and by abuse of notation just write $\mathfrak{m}$ for the maximal ideal of $A$.

Define $a_n = (x_0^{\max \lbrace n,0 \rbrace }, x_1^{\max \lbrace n-1,0 \rbrace }, x_2^{\max \lbrace n-2,0 \rbrace }, \dots, x_i^{\max \lbrace n-i,0 \rbrace }, \dots)$. That is, $a_1 = \mathfrak{m}$, $a_2 = (x_1^2,x_2,x_3,\dots ) $ and so on. Clearly $a_{n+1} \subsetneq a_n$, and $\cap_n a_n = (0)$. But also by construction for any $k>1$, no containment $a_n \subset \mathfrak{m}^k$ holds for any $n$.