User blog comment:B1mb0w/J Function Sandpit J 0/@comment-27173506-20160125061148

You have some mistakes here. f_e_0 (f_e_0 (3)) is not the same as f_w^w^w (f_w^w^w (3)), because the former one will expand to f_w^w^w^w^w...w^w (a) with a w's while the latter will be f_w^w^w (a) for some large a.