  The modulus of a polynomial-like (PL) map is an important invariant that
controls distortion of the straightening map and, hence, geometry of the
corresponding PL Julia set. Lower bounds on the modulus, called complex a
priori bounds, are known in a great variety of contexts. For any rational
function we complement this by an upper bound for moduli of PL maps in the
satellite case that depends only on the relative period and the degree of the
PL map. This rules out a priori bounds in the satellite case with unbounded
relative periods. We also apply our tools to obtain lower bounds for hyperbolic
lengths of geodesics in the infinitely renormalizable case, and to show that
moduli of annuli must converge to 0 for a sequence of arbitrary
renormalizations, under several conditions all of which are shown to be
necessary.
