= Solution
Let $Y$ be a nonconstant <Bernoulli random variable>, let $\mathcal F_t$ be trivial for $t<1$, and let $\mathcal F_t=\sigma(Y)$ for $t\geq1$. This is a right-continuous filtration after completion. The process
$$
X_t=Y\mathbf1_{[1,\infty)}(t)
$$
is adapted and càdlàg. If it were predictable, its value $X_1=Y$ at the deterministic time $1$ would be measurable with respect to the <left-limit sigma-algebra> $\mathcal F_{1-}=\sigma(\bigcup_{t<1}\mathcal F_t)$, which is trivial. That contradicts the choice of $Y$, so $X$ is not predictable.
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