Normalized partial sums of normal random variables are dense in $\mathbb{R}$

In fact we don't even have to assume that the $X_i$ are normally distributed. Just assume they have mean zero and variance $1$.

Fix some $a<b \in \Bbb R$. By reverse Fatou's Lemma we see that $$P\big( \frac{S_n}{\sqrt{n}} \in [a,b] \;\;\text{ for infinitely many $n$ } \big) \geq \limsup_{n \to \infty} P\big(\frac{S_n}{\sqrt{n}} \in [a,b] \big) = P\big(Z \in [a,b] \big)>0$$ where $Z$ is normally distributed, and the equality after the limsup follows by the central limit theorem. On the other hand, note that the event $E_{a,b} :=\big\{ \frac{S_n}{\sqrt{n}} \in [a,b] \;\;\text{ for infinitely many $n$ } \big\}$ is an exchangeable event and therefore the Hewitt-Savege 0-1 Law together with the above computation implies that $P(E_{a,b})=1$.

Finally, note that $$P\big( \{ S_n/\sqrt{n}: n\in \Bbb N\} \text{ is dense in } \Bbb R \big) = P \bigg( \bigcap_{\substack{a<b \\ a,b \in \Bbb Q}} E_{a,b} \bigg) = 1$$