I mostly have cave crystals that grow on their own on my mind, along with perhaps river channels that change their shape over time - but perhaps there are some other, better examples. Also, the more examples, the better.
There are many self-organizing, non-living complex systems. The weather is one (with the Great Red Spot on Jupiter perhaps representing its pinnacle as a self-organizing system). Snowflakes are another, and the shapes of rivers are a third.
If you are interested in this topic, then I highly recommend that you research chaos theory. James Glieck's Chaos is the canonical beginner's introductory text.
The best example is probably our universe! Current models of mass and gravity, plus some chemistry, show that the galaxies and stars that we see have "evolved" from previous forms and become quite complex indeed. See this Youtube user page for a variety of animations.