Derek Lowe has an interesting post today on In The Pipeline about common and not-so-common organic solvents. Solvation is still one of the least-understood and yet most important concepts in chemistry, as almost all synthetically useful reactions take place in solution. The dielectric constant is one way of organizing solvents based on a single property (polarity), but another possibility is to use polarizibility.
The other thing to keep in mind is that common solvents need not necessarily be reaction solvents, although that is what first comes to mind. Acetone is used by the gallon in labs all over the world for cleaning glassware, and ethyl acetate (along with hexane) is one of the most popular solvents for chromatographic separation of relatively nonpolar organics. Acetone and ethyl acetate are too reactive to be used as reaction solvents; whenever I have used them in preparative chemistry, I always used them as reagents. In my experience with synthetic chemistry, I haven’t used too many unusual solvents so far. I have used trifluoroethanol and TFA as solvents, in a very low-yielding reaction for the preparation of triphenyloxonium ions. I used a lot of trifluoromethanesulfonic acid in my research as a solvent for superelectrophilic Friedel-Crafts reactions, although this is not really practical and is mainly of academic interest, due to the high cost of triflic acid. It can be recovered after a reaction workup, but it is very laborious.
I think that this prep from Organic Syntheses for hexaphenylbenzene takes the cake for weird solvents, however.
The solvent is….refluxing benzophenone. Most compounds would decompose at such high temperatures, but if you need to really kick a recalcitrant Diels-Alder, Cope, or Claisen reaction along, then sure, break out the diphenyl ether or benzophenone!