I think non-recognized countries are probably the best best...places like South Ossetia, the previously mentioned Transnistria and Abkhazia or the Nagorno-Karabakh Republic are not extremely far from major airports but usually involve customs and passport stamps so are in no way as easy to get to as San Marino, The Vatican, etc.

Here's another fun question: which alliance airline has the most destinations in non-recognized countries? Turkish Airlines would be my candidate, with flights to Kosovo and Northern Cyprus

edit: Turkish Airlines would also be my choice for the alliance airline biggest list of "interesting" destinations, with 4(!) cities in Iraq covered, as well as Kabul...good on them for helping those communities recover!