= Solution
Planar Brownian motion is recurrent. More explicitly, the <planar Brownian annulus hitting probability> gives
$$
\mathbb P_x(T_1<T_R)
=\frac{\log R-\log|x|}{\log R}
\longrightarrow1
\qquad(R\to\infty)
$$
when $|x|>1$. Thus the unit disc is hit almost surely from every starting point. Applying the <Strong Markov property> after each departure and return shows that such returns occur after arbitrarily large times. Therefore
$$
\boxed{\{t\geq0:|X_t|\leq1\}\text{ is almost surely unbounded}.}
$$
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