The sandpile model was introduced in 1987 by physisists Bak, Tang and Wiesenfeld as a tool to study what they called the self-organized criticality—spontaneous appearance of power laws or fractal interfaces, observed in some natural phenomena. The mathematical study of the model was initiated a couple of years later by Deepak Dhar. It begins with a simple cellular automaton (also known in combinatorics under the name of chip-firing game on a finite graph, and leads to interesting long time and large volume limit dynamics when considered on increasing sequences of graphs. After an introduction to the Abelian sandpile model (ASM), I will show that the recent theory of self-similar groups (also known as automata groups) is a natural source of such families of graphs, giving rise to new interesting asymptotics and providing new evidence for the limit behaviour of the ASM.