A class of measures whose almost all points are not very well
approximable by vectors with rational coordinates, will be defined
and discussed. We prove that this class contains all Gibbs
measures for finite conformal iterated function systems and Hausdorff
measures on the limit sets of these systems (this latter is an extension
of a conjecture due to Kleinbock, Lindenstrauss and Barak Weiss).
We prove that also conformal measures of iterated function systems,
generated by the continued fraction algorithm restricted to some
(not necessarily finite) sets of entries, belong to this class.