Cabinet of emergence — specimen 05
A grid where each cell holds grains of sand. Drop one grain on a random cell; if a cell reaches 4, it topples — loses 4, gives 1 to each neighbor, which can trigger more toppling. No target height, no rule that says "stay near the edge of collapse." The question: does the pile organize itself to that edge anyway, regardless of how it started, and do the resulting avalanches have no characteristic size — some tiny, some enormous, no typical one in between?
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Watching whether average height climbs and plateaus on its own — that plateau, if it happens, is the "self-organized" part.