I was trying to find module with no maximal proper submodules (such a module can't be finitely generated). No spoilers please. So far I have failed, but in the course of looking for one I stumbled across an interesting module, formed by combining all the nontrivial prime elements together and considering the elements with finite support. This gives us something that reminds me of the product topology.
I'm curious what this module is called so I can read more about it.
Let $\mathbb{P}$ be the prime numbers.
The notation $\prod a : A \mathop. B(a)$ is called a dependent type. It is a generalization of a function $A \to B$ except that the codomain is allowed to depend on the domain, so we get something like $A \to B(a)$, except that the association between $a$ and $A$ is explicit.
$R$ is $\mathbb{Z}$ and $M$ is the subset of $\left(\prod p : \mathbb{P} \mathop. \mathbb{Z}/p \mathbb{Z} \right)$ consisting of functions where only finitely many domain elements are zero, i.e. functions with finite support.
This module has an interesting property; namely that we can use the Chinese Remainder Theorem to isolate parts of the support of each element.
Let $f$ be an element of $\left(\prod p : \mathbb{P} \mathop. \mathbb{Z}/p \mathbb{Z} \right)$ whose support is $\Delta$. For each $\mathbb{Z}/p\mathbb{Z}$ in $\Delta$ there exists a non-negative integer $k$ such that $kf$ is $1$ mod $p$ and $0$ mod every other prime in $\Delta$.
Thus, the maximal proper submodules of $M$ are precisely the submodules that are constantly zero when we restrict attention to a single prime number $p$ and consider $m_{\mathbb{Z}/p\mathbb{Z}}$ for each element $m$ of $M$.