  The R\'enyi entropies $R_{p}[\rho], 0<p<\infty$ of the probability density
$\rho_{n,l,m}(\vec{r})$ of a physical system completely characterize the
chemical and physical properties of the quantum state described by the three
integer quantum numbers $(n,l,m)$. The analytical determination of these
quantities is practically impossible up until now, even for the very few
systems where their Schr\"odinger equation is exactly solved. In this work, the
R\'enyi entropies of Rydberg (highly-excited) hydrogenic states are explicitly
calculated in terms of the quantum numbers and the parameter $p$. To do that we
use a methodology which first connects these quantities to the
$\mathcal{L}_{p}$-norms $N_{n,l}(p)$ of the Laguerre polynomials which
characterize the state's wavefunction. Then, the R\'enyi, Shannon and Tsallis
entropies of the Rydberg states are determined by calculating the asymptotics
($n\rightarrow\infty$) of these Laguerre norms. Finally, these quantities are
numerically examined in terms of the quantum numbers and the nuclear charge.
