Quoting Ludwik Kowalski <kowalskil@MAIL.MONTCLAIR.EDU>:
Many other examples of that kind can be invented,
1) Here's one that is particularly graphic:
Start with a not-too-light flexible chain or rope and hang it
across the room. You can propagage readily-visible transverse
waves along it.
Next, support the chain (at one or more points) by a thin
vertical string that is not too long or too short. For
small-amplitude horizontally-polarized waves, the string
has almost no effect; the system is almost linear. But
if the wave amplitude becomes comparable to the length of
the string, things obviously become extremely nonlinear.
2) A completely non-contrived example is sound waves in the
atmosphere. I guarantee you it becomes nonlinear whenever
the Sound Pressure Level approaches 1 Atm.
(For obvious reasons I'm using linear and nonlinear as
near-synonyms for superposition and lack thereof.)