Every field has party games. Physicists have Schroedinger’s cat. Theologists discuss the number of angels who can dance on the head of a pin. And philatelists discuss what country would be the hardest to complete and, if complete, what country would be the most valuable.
The most difficult countries to complete would obviously be those for which there are unique stamps. Sweden has an error of color of the first stamp that they issued which is unique. This stamp has hopped around many auctions in recent years indicating that in whoever’s vault it currently resides, it would only take money to pry it free. So Sweden would be very tough but not impossible. The US has a unique stamp too, the 1c Z grill (actually there are two 1c Z grills, but one is in the New York Public Library as part of the Miller collection. The New York Public Library is a library that it would be very difficult to deaccension a stamp from- it might be easier to get this stamp out of that collection if the Miller collection hadn’t had such a problem in earlier days with theft, or if the Library were part of an organization to which a million dollars or so might matter). The 1c Z grill in private hands is owned by Bill Gross and is currently not for sale for love or money. But after Gross wins all the Grand Prixs at the 2016 International show in New York, another collector may well have an opportunity to create a complete US collection. The real tough unique stamp is the 1c magenta British Guiana. It was owned by John Dupont who died in prison and the stamp is caught up in several involved legal cases including ones with the Commonwealth of Pensylvania and the estate of the man who DuPont killed. The stamp may not be sold for decades.
These three countries are hard to complete because the stamps may never be available. But how about cost? Complete US would run about $15 million, if you could do it- far more expensive than British Guiana or Sweden. For fun though, try Indian Native States. There are many very rare stamps that even with the late run up of Indian prices still sell for only a few dollars. And if you could complete that area you would have a philatelic achievement that has never been accomplished before.