Sets of "non-typical" points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes , we prove that the set of points where the pointwise dimensions, local ent ropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneous ly, carries full topological entropy and full Hausdorff dimension. This fol lows from a much stronger statement formulated for a class of symbolic dyna mical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.