score:16
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.
Upvote:1
Upvote:2
Cešký Těšin, the western part of Cieszyn, Silesia:
Upvote:6
Lviv:
This does not qualify as the most number of different countries, but probably qualifies as the most number of changes in national flag.
Upvote:20