Simple example for a predictable process that is not in $L(X)$ for a semimartingale $X$

Question

I am supposed to find some simple and basic examples for semimartingales $X$ and predictable processes $H$ such that $H\notin L(X)$, that means $H$ is not integrable with respect to $X$. Do you have such examples in mind? Are there even processes which are commonly used in finance / physics that can be used as examples?

Thank you very much.