  For a dynamical system, we study the set of points $\cal W$ whose orbit
approximates any chosen point at certain specified rates. Our basic setting is
that of left shift acting on topological Markov chains endowed with a local
weak Gibbs measure. Our rates of recurrence are so fast that the corresponding
set $\cal W$ has measure zero, but we obtain a generalized Bowen formula for
Carath\'eodory dimension. For the case of Markov transformations with countable
partition and big image (BI) property a Bowen-type formula is obtained for the
Hausdorff dimension of those exceptional sets. In particular, we apply our
general results to Gauss and Luroth maps, a not Bernoulli modification of Gauss
map and some inner functions. Since we only require the existence of weak Gibbs
measures we can deal with non H\"older potentials and we can also consider
intermittent systems as the Manneville-Pomeau map.
