Dense uncountable proper subgroup of $(\mathbb{R},+)$
Here's one from Hardy's "A course of pure math":
$\displaystyle \{x \in \mathbb{R}: \lim_{n \to \infty} \sin(n! \pi x) = 0\}$.
Quoting an answer of François Dorais on Mathoverflow:
This earlier answer of mine shows how to get an uncountable $\mathbb{Q}$-independent subset of $\mathbb{R}$ in ZF. This set is not a Hamel basis so the $\mathbb{Q}$-span of this set is as required.