= Solution
A simple example is
$$
\boxed{M_t=B_t^2-t}
$$
for a standard <Brownian motion> $B$. The <Itô formula> gives $M_t=2\int_0^tB_s\,dB_s$, a continuous square-integrable <martingale> on every finite horizon. For $h>0$, write $\Delta=B_{s+h}-B_s$, independent of the past and normal with variance $h$. Then
$$
M_{s+h}-M_s=2B_s\Delta+\Delta^2-h,
\qquad
\mathbb E[(M_{s+h}-M_s)^2\mid\mathcal F_s]=4B_s^2h+2h^2.
$$
Since $B_s^2=M_s+s$, this conditional second moment is already a nonconstant function of the <martingale>'s own past for $s>0$. The <tower property of conditional expectation> gives the same expression on conditioning on that own past. Were the increments independent, that conditional second moment would be constant. Thus the increments are not independent.
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