In principle, I think there is an infinite number.
In principle, an electron could enter a positive Ion and move down the energy levels one at a time! It is unlikely, however, as it is likely to take larger steps.
Google says:
The equation below defines the formula for determining the energy levels of Hydrogen atoms.
(And n = 1,2,3,…, and so forth. Because it takes so much energy to remove an electron from a nucleus (or unbind) it, the energy is expressed as negative numbers.
As you can see, the levels are very close together.