I am not sure it is safe to say "it is not possible" to add or remove more than three electrons from an atom. For example you can have the compound
Tin(IV) phosphate. In this form, the tin has a +4 ionic charge and therefore has loss 4 electrons.
I think it would be better to say that representative elements tend to lose electrons that move down to the closest noble gas configuration (metals) or gain electrons to move up to the next closest noble gas (nonmetals). For most representative elements, due to their location in the periodic table, do not need to gain or lose more than three electrons to accomplish this.
One could also argue that the energy required to remove (or add) each subsequent electron is more than the previous one and eventually the energy required to remove the electrons are not compensated for by the energy released when forming the new ionic compound and thus for energy reasons we do not see the loss, or gain, of more than three electrons for well behaved representative elements.
First of all, with enough energy you can remove any number of electrons. In the sun's interior for example, the temperature is so high that the atoms are all in the plasma form with the electrons separated from the nuclei. Getting back to your question though...
When you remove an electron from a neutral atom, it will have a net positive charge. As you remove more electrons, this charge increases. This positive charge attracts electrons, so that if you take too many off, it will pull them right back on!
It deals with stability and/or the energy required to take away the electrons.
When an atom meets the "octet rule" it does not want to gain or lose electrons.