"What is the largest market in the world still operating as a barter economy?"
For the solution, scroll down to the bottom of the page.
At a cool 2 trillion or so a day of volume, the answer is: the foreign exchange market. Currency pairs are traded at an exchange rate without the use of a third good acting as a common denominator. If you want to be picky (like one of my professors...) you can say that the weird pairs like Baht/Ruble, usually go through either the Euro or the Dollar. These pairs account for a negligible amount of volume, so in any case what's left would still be the largest barter market in the world.
Nobody guessed this so far, because as soon as you mention barter, people start thinking third world countries or stuff like that. To be fair I don't think I would have guessed it either, I was thinking about barter markets in general and wondering how an economy would have to work without currency, and then thought "oh it would be like the foreign exchange". From there I made up the question.
Anyway I think it's pretty cool that the largest market (by volume) in the world uses such a "primitive" market structure.