One of the best non-computer projects I know for observing chaos is to build a double pendulum -- a pendulum suspended from another pendulum. The motion of the double pendulum is quite complicated. The second arm of the pendulum sometimes seems to dance about under its own will, occasionally executing graceful pirouettes while at other times doing a wild tarantella.
You can make the double pendulum from wood. At the pivot points, you might try to use ball bearings to insure low friction. (Ball bearings can be obtained from hobby shops or from discarded motors and toys.) Place a lead weight at the bottom of the first pendulum so that the pendulum will swing for a longer time. (The weight stores potential energy when the pendulum is lifted.) The second pendulum arm can be about half the length of the first.
You can place a bright red dot, or even a light, on one end of the second pendulum so that your eye can better track its motion. Note that your pendulum will never trace the same path twice. This is because you can never precisely reposition it at the same starting location due to slight inaccuracies in knowing where the starting point is. These small initial differences in position are magnified through time until the pendulum's motion and position becomes unpredictable.
Can you predict where the lower pendulum will be after two or three swings? Could the most powerful supercomputer in the world predict the position of the pendulum after 30 seconds, even if the computer were given the pendulum's precise equations of motion? Unlike the strange attractor patterns in books (or on web pages), your pendulum's pattern will eventually come to rest at a point due to friction.