You can use the fact that $B_t-m_t$ is a reflected Brownian motion (see e.g. Revuz-Yor, Chapter VI, Theorem 2.3). I think it shouldn't be difficult to show that
$$
\limsup \frac{M_t-m_t}{\sqrt{t\log\log t}} = \limsup \frac{B_t-m_t}{\sqrt{t\log\log t}}.
$$