Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. More generally semimartingales with independent increments are Markov. I'm interested in examples of popular classes of processes that are outside of this intersection. For instance, Hawkes processes are semimartingales, but are not Markov. Are there other such examples? Vice versa, what are interesting specific examples of Markov processes that are not semimartingales?

Update: googling lead me to the following paper: "Stochastic calculus for symmetric Markov processes" by Chen, Fitzsimmons, Kuwae and Zhang, http://projecteuclid.org/euclid.aop/1207749086

If $(B_t)_{t \geq 0}$ is a Brownian motion, then $(|B_t|^{\frac{1}{3}})_{t \geq 0}$ is not a semimartingale (see this question) but a Markov process.