I stipulate that a territory "changes hands" when a polity actually administers it. For example, territory which is briefly controlled by an invading military force but not administered doesn't count.
Edit: I meant to limit the scope of my question to the 20th century.
Edirne in Turkey has been identified as "the most contested spot on the globe," and indeed I believe that the Rumelia/Macedonia region is a good candidate for this.
Well, that's hard to beat but I think that both Istria and Dalmatia would make a fine example. These lands were, and still are, home to various ethnic groups - mainly Italians (most of whom were forced to flee or were murdered from 1943-49) and Slovene/Croatians, but there were also small pockets of Jewish, Hungarian, Serb and Bosniak minorities.
Until 1797, these lands were ruled by Venice; Napoleon then gave them away to the Austrians, who kept both of them as part of the Littoral province until 1919, when they were split between the Kingdom of Italy (the Julian Marches, Istria as well as Zara and its environs) and The Kingdom of Serbs, Croats and Slovenes; borders were further amended with the 1924 Treaty of Rome.
After 1943, Nazi Germany annexed both of them as part of the Adriatic Operation Zone; in 1945 these were divided by the Allies, who established two zones (A and B) of occupation separated by the Morgan Line. Trieste was declared a "Free Territory" under U.N. protection.
In 1947, Yugoslavia annexed Zone B as well as most of Zone A; the rest of it and the Free Territory of Trieste were reunited with Italy after the London Memorandum of 1954, and have been part of it ever since.
The borders were further amended in 1975 with the Treaty of Osimo.
After 1991, most of the Julian Marches and part of Istria belong to Slovenia; the rest and Dalmatia, to Croatia.