I was thinking about the following question:
Find a non-Noetherian module which is not a ring.
All the examples I have in mind can be thought as rings. For instance, one can take the space of real sequences $(a_n)_{n \in \mathbb{N}} \subset \mathbb{R}$. It is indeed an $\mathbb{R}$-vector space which is infinite dimensional and hence not Noetherian. However it is also a ring, and it is not Noetherian also as a ring.
I was thinking of something similar to this: an infinite-dimensional vector space $V$ over a field $\mathbb{k}$, but where I cannot put the structure of a ring. I tried spaces of functions like $\mathbb{C}^{\infty}(\mathbb{R}^n)$ or Lebesgue spaces of functions $L^p(\mathbb{R}^n)$, but here too, since I can multiply functions, it still makes this a ring.
So as it turns out, every set $S$ can be thought as a ring. Just consider: $$ \bigoplus_{s \in S} \mathbb{Z} $$
So the only possible answer to the question is to find a set which is only equipped with a module structure which is not a ring structure and such that it is not Noetherian. As already pointed out by myself, one can search within infinite dimensional vector spaces. For instance, as suggested by @Mark in the answers, if $\mathbb{k}$ is a field, $\mathbb{k}[x]$ is an infinite dimensional vector space, which is hence not Noetherian.