Even educated non-scientists know about fullerenes, the novel allotropes of elemental carbon resembling soccer balls or geodesic domes. C60, a spherical structure containing 60 carbon atoms, is the archetype. Each carbon atom in a classical fullerene is sp2 hybridized, meaning essentially that it is bound to three other atoms arranged more-or-less in a plane with it. A carbon atom is said to be "saturated" if it has four bonds (sp3 hybridization), and any carbon with less than four--like those in fullerene structures--is said to be "unsaturated" because it could, at least in terms of classical valence bond theory, accept at least one more bond. These are the same "saturated" and "unsaturated" that gives us the terms "saturated fat" and "unsaturated fat." The chemistry to take an unsaturated carbon to a saturated one is rudimentary and is practiced every day on vast industrial scales.
So I'm driving along the other day and it occurs to me that fullerenes are unsaturated--they're just carbon. Could we dump them in a reactor with hydrogen and a metal catalyst, just like we do with the vegetable oil that ends up in your oreo cookie filling, and produce the saturated hydrocarbon equivalents of fullerenes? Hydrofullerenes? So I went to the library and, per Hirsch and Brettreich's excellent book Fullerenes, found out that the short answer is "Yes, but not exhaustively." While partially-hydrogenated fullerenes like C60H36 can be produced and are relatively stable, exhaustive hydrogenation has not been achieved and is probably impossible, at least under practical conditions. This is believed to be a consequence of steric crowding on the exterior of the carbon shell; the more positively-charged protons you stick on to it, after a point, the less stable it gets.
The next-most intuitive question, at least for me, is "How about fluoridation?" The realization that flourine atoms can be treated analogously to hydrogen atoms in hydrocarbon chemistry gave us Teflon and the whole modern field of fluorocarbon chemistry. So if we can't make perhydrofullerenes, how about their perfluoro analogs? A sort of "Teflon sphere" idea? Turns out, again per Hirsch and Brettreich, that the answer is "No." Again, while partially-fluorinated fullerenes can be and have been produced, perfluorination turns out to be unfavorable for reasons which are analogous to those which disfavor perhydrogenation. The only difference is a sign change: While the surface of perhydrofullerene is too positively charged to be stable under practical conditions, the surface of perfluorofullerene is too negatively charged to be stable under practical conditions.
So my hare-brained idea is this: Try to fully saturate C60 using a "hetero-fluoro-hydro" strategy, so that the complimentary positive and negative partial charges of protons and fluorine atoms on the sphere's surface stabilize the structure. You could either hydrogenate and then fluoridate, or fluoridate and then hydrogenate. My intuition favors the latter, because while it's known that fluorine will displace hydrogen, the opposite reaction does not occur, to my knowledge.
I'm not an expert in the field by any means but I've done some rudimentary literature searches using phrases like "hydrofluorofullerene," "fluorohydrofullere," etc. and not found any precedent.
As to benefit, who knows? My readings to date indicate that fully saturated fullerenes of any type have been produced only in trace quantities, if at all. It would be a significant achievement to produce saturated fullerene in significant yield. Then you study its properties and start to think applications. If nothing else, being the first to make lots of saturated C60 could be good for one's scientific career.
The point has been made that the fullerenes and fullerene type structures are highly stable. They are even more stable, in fact, than carbon in its adamantane geometry (i.e. diamond), because the sp2 hybridization of the carbon atoms in fullerenes allows for an enormous amount of resonance stabilization when the double bond electrons delocalize through the enormous pi-system. (Which is what makes them conductive.) This is something I glossed over earlier in discussing the energy costs of saturating fullerenes, when I only mentioned steric repulsion at the surface. If you saturate a fullerene, you're also breaking a very large resonance stabilization. This is why, as some have suggested, it appears to be feasible to exhaustively perfluoridate diamond surfaces--adamantane carbon is sp3. But it is not safe to assume that because diamond can be perfluoridated, so can fullerenes, again because diamond is not resonance stabilized and fullerenes are.
The hetero-fluoro-hydro strategy I propose might offset the steric costs of saturation with complimentary electrostatic interactions on the surface, but I don't think it'll help much with the resonance-breaking problem. However, because the studies I've seen suggest that it's really not too hard to at least partially saturate C60, my intuition is that the steric problem is much more significant than the resonance-breaking problem. After all, the first double bond should be the hardest to break, because it will have the most extended resonance and hence the most stabilization. And since they've already made it to C60H36 by conventional hydrogenation techniques, it follows that sterics are the limiting factor, not resonance.
Peroxidation appears to be a workable strategy, c.f. Chemical Physics Letters 384 (2004) 283-287. Tsukuda and co-workers demonstrate convincingly that they can produce C60On with n <= 30 by corona discharge ionization. Again, it hasn't been done in quantity, but Hirsch and Brettreich seem to think it could be. The paper includes a really cool figure showing C60O30. I would also note that traditional "wet" metal catalytic epoxidation has been tried many ways, and they can't seem to get more than 6 oxygen atoms installed.