No iterated identities satisfied by all finite groups

We show that there is no iterated identity satisfied by all finite groups. For w being a non-trivial word of length l, we show that there exists a finite group G of cardinality at most exp(l^C) which does not satisfy the iterated identity w. We also prove a more general statement concerning iterations of an endomorphism of a free group. The proof uses the approach of Borisov and Sapir, who used dynamics of polynomial mappings for the proof of non-residual finiteness of some groups.