  It is known that the behavior of the Mitchell order substantially changes at
the level of rank-to-rank extenders, as it ceases to be well-founded. While the
possible partial order structure of the Mitchell order below rank-to-rank
extenders is considered to be well understood, little is known about the
structure in the ill-founded case. The purpose of the paper is to make a first
step in understanding this case, by studying the extent to which the Mitchell
order can be ill-founded. Our main results are (i) in the presence of a
rank-to-rank extender there is a transitive Mitchell order decreasing sequence
of extenders of any countable length, and (ii) there is no such sequence of
length $\omega_1$.
