As you increase the energy level of a hydrogen atom $n \rightarrow \infty$ I have learned that the energy of that energy level changes according to
$$ E_n =- \frac{13.6 {\rm eV}}{n^2},$$
and that there are technically an infinite set of these energy levels. As you get closer to $n=\infty$ the energy appears to decrease more and more rapidly, as it gets closer and closer to 0, at which point the the electron would no longer bound in quantized orbits of angular momentum. The angular momentum is given in the Bohr model as
$$ L = n\hbar,$$
and if you set this equal to an other quantity like $mrv$ or $r\times p$ you could get a radius for the orbit $r$. It is taught that this radius grows and technically also goes to infinity, is this always true? Is there any modifications that make the orbits converge at a certain distance as in something like a Poincare disk (see image)?