Consider the ring $C([0,1])$ of continuous functions on the interval $[0,1]$ with pointwise addition and multiplication. Let $X_n=[0,\frac{1}{n}]$, and ${\mathfrak{a}}_{\mathfrak{n}}=\{f\in C([0,1])|f(x)=0,x\in X_n\}$. This gives us a strict chain ${\mathfrak{a}}_{\mathfrak{1}}\subset {\mathfrak{a}}_{\mathfrak{2}}\subset {\mathfrak{a}}_3\subset \dots$ which does not terminate. This is a nontrivial example of a 1;;non-noetherian ring.

Typically holomorphic functions and polynomials are more well behaved, with their rings being Noetherian because of more strict conditions.