The double pendulum. A good example of "sensitivity to initial conditions" (i.e. Chaos). The pendulums start in very slightly different state (so slight it's not even detectable by eye). For some time, their behavior is the same, but then it begins to diverge. The interesting thing about this phenomena is that, even for mathematically predictable phenomena, our control over initial conditions will always be limited. A well known example is, of course, the weather - for which predictions about the next day or two can be pretty accurate, but predictions about a week away will likely be updated as the system diverges from prediction and meteorologists re-calculate predictions based on the new state.
