  An automorphism $F$ of the polynomial ring in $n$ variables over a field of
characteristic zero is said to be {\it co-tame} if the subgroup of the
automorphism group of the polynomial ring generated by $F$ and affine
automorphisms contains the tame subgroup. There exist many examples of such an
$F$, and several sufficient conditions for co-tameness are already known. In
2015, Edo-Lewis gave the first example of a non-cotame automorphism, which is a
tame automorphism of the polynomial ring in three variables. In this paper, we
give the first example of a non-cotame automorphism which is not tame. We
construct such an example when $n=3$ as the exponential automorphism of a
locally nilpotent derivation of rank three.
