Metohija Ottoman Empire (until 1912) Montenegro (1912-1915) Austria-Hungary (1915-1918) Serbia (1918) Kingdom of Serbs, Croatsand Slovenes (1918–1929) Kingdom of Yugoslavia (1929–1941) Italian protectorate of Albania (1941-1943) Nazi Germany (1943-1944) Serbia (1944-1946)
Yugoslavia (1946-1991) Federal Republic of Yugoslavia(1992-2003) Serbia and Montenegro (2003-2006) Serbia (2006-2008) Kosovo (2008-
history.stackexchange.com/users/33710/user33710
I think the Carpathian Ruthenia area (the little triangle on the "end" of Czechoslovakia after the end of WW1) gives us five bona fide separate national entities and more if pseudo-states (such as your Fiume) and other periods of loose or brief control are allowed.
up to 1918 Austria-Hungary 1918 Hungary 1918
West Ukraine 1918-1939
Czechoslovakia 1939 Carpatho-Ukraine Republic 1939-1944 Hungary 1944-1945 Czechoslovakia 1945-1991 USSR
1991 - Ukraine
I've marked in bold where nation states were recognisably exercising full sovereignty over the area.
Looser claims in italics.
Memelland (now Klaipeda) in Lithuania is another possible candidate.
history.stackexchange.com/users/106/Tea Drinker
Lviv : Austria-Hungary (until 1918) Western Ukrainian National Republic Poland (1918-1939) USSR (1939-1941)
Reichskommissariat Ukraine (1941-1944) Poland (1944-1946) USSR
(1946-1991) Ukraine (1991-now) This does not qualify as the most number of different countries, but probably qualifies as the most number of changes in national flag.
history.stackexchange.com/users/1979/sds
Cešký Těšin , the western part of Cieszyn, Silesia : to 1918 : Austria-Hungary October 31-November 5 1918: competing Czech and Polish local administrations November 5
1918- Jan 30 1919:
Polish occupation 1919-1920:
Czech occupation 1920-1938:
First Czechoslovak Republic (formalized by 1920 Spa Conference) 1938-1939: Second Polish Republic (Polish occupation after Munich Agreement) 1939-1945: Großdeutsches Reich (i.e. annexed by Nazi Germany) 1945-1992: Czechoslovakia 1993-: Czech Republic/Czechia
history.stackexchange.com/users/21946/Spencer
U Sabotů , now Šance (see the map ). until 1918: Austria(-Hungary) 1918
- 1939 Czechoslovakia 1939 - 1945 Germany (Protektorat Böhmen und Mähren) 1945 - 1992
Czechoslovakia 1993 - 1997 Czech Republic 1997 - Slovakia
history.stackexchange.com/users/21523/Radovan Garabík
For example, a centenarian who has spent her entire life in  Rijeka  would have lived in five:   The Austro-Hungarian Empire (until 1920)  The Free State of Fiume (1920-1924)  Italy (1924-1947)  Yugoslavia (1947-1991)  Croatia (1991-present)   Is there any piece of land in the world which has been in more hands than that since March 5, 1898 (the birthdate of the  oldest living person )?  If so, where?
How many different countries could a person alive today have lived in without leaving their hometown?
history.stackexchange.com
<20th-century><geography>
history.stackexchange.com/questions/10275
history.stackexchange.com/users/1003/dan04
The Carpathian Ruthenia area and Memelland in Lithuania have also been part of many different states over the last century. The regions of Carpathian Ruthenia and Memelland could potentially have changed hands more than five times since March 5 1898.
The regions of Carpathian Ruthenia and Memelland could potentially have changed hands more than five times since March 5 1898.