In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each q>0, zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system (X,T) (where X=Mℤ is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q>1, infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s∈(0,1) and each q>1, zero lower s-generalized and infinite upper q-generalized dimensions.