Re: how many energy levels does a hydrogen atom have?

The energies E of the hydrogen levels are given in electron 
volts (eV) by the formula E=-13.6 / n^2, where n^2 means n squared, and 
n is any non-zero positive integer: 1, 2, 3, 4, ... 

As you see, there is an infinite number of levels, yet they 
are all within the interval -13.6 eV to 0 eV. 

The electron can be excited to any of these levels, in a 
single jump or several jumps through intermediate levels. 
The electron can also be excited to a positive energy to 
become free from the atom if it absorbs enough energy. This 
is called "ionization". Thus if the electron is in the lowest 
state: n=1 (E=-13.6 eV), and it receives an energy greater 
than 13.6 eV, the atom will be ionized. Of course if the 
electron is in state, say, n=5, we need at least 13.6/25=0.54 eV 
to ionize the atom. The energy needed to set an electron 
free from the atom is called the binding energy. 

The above applies to an isolated atom. The 
radius of the orbit of an electron in state n in amstrongs 
is 0.529 n^2 Since n can be arbitrarily large, the orbit 
can be as large as we want, and this is unrealistic because 
atoms are usually surrounded by other atoms or particles 
that are colliding with it all the time.  
As n grows, the binding energy of the electron becomes weaker. 
At some point those collisions with its 
surroundings will give the electron enough energy to overcome its 
binding energy and ionize the atom. This means that no atom can 
really have an infinite number of levels. The number of effective 
levels that an atom has depends on how far other atoms are, 
that is, it depends on the density of the medium. In interstellar 
space, atoms can have several hundred levels, while in the 
atmosphere atoms only have a few n states.