= Solution
Write $N_t=M_t-M_0$ and $V_t=\langle M\rangle_t=[N]_t$. The <Dambis-Dubins-Schwarz theorem> supplies a standard <Brownian motion> $W$ such that
$$
N_t=W_{V_t}.
$$
On the event $V_\infty=\infty$, continuity and monotonicity imply that the range of $t\mapsto V_t$ is all of $[0,\infty)$.
For completeness, <Brownian motion> hits every positive integer <almost surely>. The <Brownian reflection principle> gives, for $a>0$,
$$
\mathbb P\!\left(\sup_{0\leq u\leq R}W_u\geq a\right)
=2\mathbb P(W_R\geq a)\longrightarrow1\qquad(R\to\infty).
$$
Intersect over integer $a$. The times of these increasingly large values must escape every compact interval because each Brownian path is continuous there. Therefore $\limsup_{u\to\infty}W_u=\infty$. The unbounded clock transfers these values to arbitrarily late times for $N$, and adding the finite initial value $M_0$ preserves unboundedness. Hence
$$
\boxed{\limsup_{t\to\infty}|M_t|=\infty\quad\text{almost surely}.}
$$
This is an instance of <infinite quadratic variation forces unbounded oscillations>; applying the same argument to $-W$ also gives negative oscillations.
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