Examples demonstrating that the finitely generated hypothesis in Nakayama's lemma is necessary
Solution 1:
If $(R,\mathfrak m)$ is a local domain which is not a field, then for its fraction field $K$ we have $\mathfrak m K=K$.
Amusingly, this proves that $K$ is not a finitely generated $R$-module.
Solution 2:
My favorite example is $R = \mathbf Z_{(p)}$ [which is very Noetherian] and $M = \mathbf Q$. Of course, you then think of localizing $R = k[x]$.
On that note, there was a thread on MO about understanding the lemma and Roy Smith's answer there is pretty geometric: "It's sort of like the inverse function theorem".