Francium is the largest element known to man, but it does not contain all of it's possible electrons. Would there be any way to make the element receive another electron?

If it is possible to make pure Francium, I would say yes. With the technology that we have, we should be able to do so. However, what would be the reason to make pure Francium? It would certainly cost much more and a Francium of purity 90% would be sufficient for most purposes. It really depends on what you want to do with the Francium.

Yes, I don't think that it's impossible to get electrons into the empty orbitals of Francium but it would become highly unstable as you try to put more and more electrons.

Mixing any Alkaline metal with other elements cause violent reactions, but the Metals must have there entire orbit of electrons filled. If you could take pure francium with a full electron orbit, and mix it with fluorine the least filled element...

It's cause a very violent reaction that would produce a lot of energy.

If you are thinking about reacting Francium to make energy, I would say that it cannot be done. Francium is a very reactive metal and also radioactive. Francium is extremely rare in nature and the more time goes, the less remains.

Now, even if you find it in nature, it will be in the form of ion, most commonly as a singly cation (one less electron).

Now, if you want to have pure Francium, electrolysis is the way to go. But the process requires a lot of electricity and the process never 100% efficient. So much so that in the end, you'll spend more energy in making pure francium than getting energy from it.

If you want to make Francium, the same thing happens. You will need more energy to make it that energy it gives off in reactions.

I don't believe that radioactivity and energy cost will be a problem, but what would electrolysis do to fill Francium's outer orbit?

No, electrolysis will merely convert the Fr^+ ion into Fr atoms.

I would expect it theoretically possible to add an electron to Fr to make an Fr- ion. All the other alkali metals have an electron affinity and can accept an electron - why shouldn't Francium.. (Their outer s orbital is only half-filled and thus can accept a second electron.. The alkaline earth metals, with FILLED s orbitals, will not accept an additional electron. It would have to start filling the p orbitals, and it won't.)
Na- exists, so should Fr-!

What I was under the immpresion of is that the reason Francium cannot accept another electron, is because there is no other element willing to give one of it's electrons bad enough that francium would accept it. If an element could be found, wouldn't it be possible?

Put enough voltage behind it and YOU can accept an electron! As Unknown008 said, electrolysis has been the traditional method of getting active metals in a pure state.

About the putting a potential difference, I'm not quite sure...

For one, I know that in the metallic state itself, metals exist as cations in a sea of delocalised electrons, meaning that the metal already 'threw away' it's outermost electron. As such, when subjected to a potential difference, I think that it would get charged without necessarily filling the orbitals of the francium cations and at a certain point (like most charged metals), the francium will just discharge the excess electrons as a spark.

The Fr^- ion would then exist most probably in the gas state than in aqueous, molten or solid state.

If we now ionise vapour Fr (g) (by using beta emission)... that could work... (though could be dangerous as Fr is very reactive and radioactive). Even then, the Fr will have to accept the electron, we cannot force it, and as such, I believe there is a limit to the number of electrons that it can accept. As I said earlier, if it had more electrons, it would be more unstable with each additional electron and completing it's orbitals seem hardly possible.