Abelian sandpile
OK so, every cell holds some sand. Drop a grain, and when a cell gets to four it tips over, one grain to each neighbor, and those neighbors might tip over too. Most drops do nothing at all; every so often one grain sets off half the grid. Nobody tuned it to do that, the pile drifts there on its own and then sits right at the edge (physicists call this self-organized criticality, which is a long way of saying it keeps almost falling over). The histogram is avalanche sizes on a log-log scale, and give it a while (or crank the grains per frame) and it straightens out into a line. Click the grid to drop grains yourself.
Abelian sandpile: drop a grain, and any cell reaching 4 topples one grain to each neighbor, cascading until stable. The avalanche sizes go power-law without anyone tuning them there. -> petri