Andrezj Zuk's [notes on Automata groups][1] gives a nice introduction to the topic computing explicit examples. In particular, his description of the Aleshin group (a finitely generated torsion group) and proof of intermediate growth is quite accessible and somewhat simpler than the proof for Grigorchuk's group. 


  [1]: https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCUQFjAA&url=http%3A%2F%2Fcimpa-icpam.org%2Farchivesecoles%2F20101208101638%2Fnotes.pdf&ei=IjaeVcC5NczqUry-tpgL&usg=AFQjCNH1zrsYVqYtYAFxWXEejSOQvj9RUA&sig2=EFR8uD-hnHnmA1t7u1IuUQ&bvm=bv.96952980,d.d24