  In this work, we are interested in characterizing typical (generic)
dimensional properties of invariant measures associated with the full-shift
system, $T$, in a product space whose alphabet is a countable set. More
specifically, we show that the set of invariant measures with infinite packing
dimension equal to infinity is a dense $G_\delta$ subset of $\mathcal{M}(T)$,
the space of $T$-invariant measures endowed with the weak topology, where the
alphabet $M$ is a countable Polish metric space. We also show that the set of
invariant measures with upper $q$-generalized fractal dimension (with $q>1$)
equal to infinity is a dense $G_\delta$ subset of $\mathcal{M}(T)$, where the
alphabet $M$ is a countable compact metric space. This improves the results
obtained by Carvalho and Condori in \cite{AS} and \cite{AS2}, respectively.
Furthermore, we discuss the dynamical consequences of such results, regarding
the upper recurrence rates and upper quantitative waiting time indicator for
typical orbits, and how the fractal dimensions of invariant measures and such
dynamical quantities behave under an $\alpha$-H\"older conjugation.
