Why doesn't helium have a third ionization energy?

Simply put, because it does not have three electrons.

is a measure of how much energy is needed to remove 1 mole of electrons from 1 mole of atoms in the gaseous state.

Simply put, ionization energy tells you how much energy is needed to pluck an electron from an atom.

This implies that an atom cannot have more ionization energies than it does electrons.

In your case, helium's first ionization energy is the energy needed to go from one mole of helium atoms to one mole of ##"He"^(+)## cations by removing one of the two electrons from the atoms in the gaseous state.

##"He " + " energy" -> "He"^(+) + "e"^(-)##

Once you do this ,each helium atom will be left with one electron. This is where the second ionization energy comes in. This time, you need to provide enough energy to remove that last electron from each ##"He"^(+)## cation to make a ##"He"^(2+)## cation, which essentially is a helium nucleus.

##"He"^(+) + "more energy" -> "He"^(2+) + "e"^(-)##

And this is where the process stops. Helium does not have a third ionization energy because there are no more electrons to remove from the atom.

So, as a conclusion, the number of ionization energies cannot exceed the number of electrons.