This iterative fractal program generates fractals like the famous Koch snowflake. You have a path, and each segment of the path then gets the same path drawn on it. It's hard to explain, but fun to play with. The app can save images of things you particularly like; hit the + button to save images, and the i button to mess with settings.

You want the ability to save and restore settings? Me too, but I haven't gotten to it yet.

Touch points on the primary path to drag them and move them. Tap the middle of a segment to add a new point. Double-tap a point to delete it. Touch anywhere else to drag, zoom works as normal, and double-tap anywhere else does its best to auto-zoom/auto-center. Shake to randomize (and reset a few settings that are easy to get lost in).