Let $Y_1, Y_2, Y_3,\ldots,Y_n,\ldots$ be iid and bounded random variables with $E[Y_1]=0$. Define $X_n = Y_1+Y_2+ \cdots + Y_n$.
If $\Pr(Y_1 \neq 0) \gt 0$, then $ \limsup X_n = \infty$ with probability $1$.
The conclusion seems intuitive, but how would I approach it rigorously?
Suppose that $ \limsup X_n = b$ for some $b \in \mathbb{R}$. We know that $|Y_i| \leq c$ for some $c \in \mathbb{R}^{+}$. So $X_n \leq nc$. Now for some large $N$, $n \geq N$ means that $X_n \leq b$. But we can choose a large enough $n$ (say $m$) such that $b < X_m \leq mc$.
Note: This is just my attempt. Please feel free to continue OP.
