It's easy to establish that any caglad process is locally bounded. However, I doubt if the same conclusion holds for cadlag process.
Let's consider a cadlag process $X$ (say it's RCLL for all $\omega$ for simplicity). Naturally, I think of the stopping times $T_n = \inf\{t:|X_t|>n\}$. It's easy to tell that for any $\omega$ $$ \sup_{0 \leq s < T_n(\omega)} |X_s(\omega)| \leq n $$ But we don't know if the (absolute) jump size at $T_n(\omega)$ is too large to keep $X(\omega)$ bounded by $n$. Indeed my doubt results from this concern.
Then I start to think about counterexample. What I have in mind is such a "weird" process: $$ X_t(\omega) = \begin{cases} 0, t<1\\ J(\omega), t \geq 1 \end{cases} $$ where $J$ is a random variable such that $P(J = 0) = P(J = \infty) = \frac 12$. It seems that $X$ cannot be localized to be bounded. But I don't fancy this example: as $X$ is deliberately allowed to take $\infty$. Whether or not it can be considered as "cadlag", it doesn't seem to be a good example.
So my question is: is there any example in which $X$ is cadlag, almost surely finite (given $\omega$, $X_{t}(\omega)$ is finite for all $t<\infty$), yet not locally bounded?