For example, imagine that I excite an electron in order to move it to the 19th energy level (when the lowest possible is, let's say, the 2nd (there are three electrons in total in the atom)). How can I know if the electron will tend to move to that n=2 level or if it will leave the atom? What is the highest energy level in which an electron would still tend to move to the lowest state possible?
Thank you in advance. You can add papers, articles etc, as I'm very interested in this topic.
In principle, there is an infinite number of excited states where you can put an electron, whose energies approach the ionization potential to arbitrarily close distances, but which are still bound to the atom; these are known as the Rydberg series of the system.
As a practical matter, of course, if you want to climb arbitrarily high up the ladder, then you need to be prepared to foot the bill for a sufficiently expensive experiment: the coupling to high-$n$ Rydberg states is very weak, so the transition probabilities will be small, and the energy spacing is very small, which means that your experiment will take a very long time to complete, and you will need a very stable setup. (In practical experiments, you normally go up to several hundred - example.) But in principle you can climb as high as you want up that infinite ladder.