Another very easy to understand is language. Say, I give you a small text. You could create a small state machine containing the possible transitions between words. You could also compute probabilities (estimated from the text) of going from one word to another (e.g the -> text vs. the -> possible, etc), and you'll have a Markov chain.

Of course, this is a very weak model of language, and usually in such models probabilities are modelled on at least the previous two states. But it turns out to be very useful in practice, e.g. you can compute the fluency of different formulations of the same semantics.