  We prove that if $J$ is the limit set of an irreducible conformal iterated
function system (with either finite or countably infinite alphabet), then the
badly approximable vectors form a set of full Hausdorff dimension in $J$. The
same is true if $J$ is the radial Julia set of an irreducible meromorphic
function (either rational or transcendental). The method of proof is to find
subsets of $J$ that support absolutely friendly and Ahlfors regular measures of
large dimension. In the appendix to this paper, we answer a question of
Broderick, Kleinbock, Reich, Weiss, and the second-named author ('12) by
showing that every hyperplane diffuse set supports an absolutely decaying
measure.
