This was an excellent question Jasen!
I went back and did some research and found that I have seen multiple versions of over 40 stories!
The only constant I see is that I always like the original movie better if the remake is done in another country. As a result Hosenhaus, this makes me more of a fan of Incompreso than of Misunderstood. It's been many years since I've seen either version although I will say that I remember the ending of the Italian film having more impact than the American one.
For the record Jasen here are the stories which I have seen multiple versions....
Stories I like the original film bestLes Comperes/Father's Day
Ransom
Cheaper By the Dozen
Incompreso/Misunderstood
Adventures of Huckleberry Finn
Adventures of Penrod & Sam
Tuck Everlasting
Spoorloos/Vanishing
The Getaway
Hansel & Gretel
Le grand chemin/Paradise
Flipper
Miracle on 34th Street
The Miracle Worker
Klatretøsen/Catch That Kid
The Hideaways/From The Mixed Up Files of Mrs. Basil E. Frankweiler
Un indien dans la ville/Jungle 2 Jungle
The Private War of Major Benson/Major Payne
Pollyanna
The Prince and The Pauper (I've seen 1962 & 2001 versions)
The best examples of movies I liked WAY WAY MORE than their remakes would be The Biscuit Eater, The Bad News Bears and Village of the Damned.
Stories I like the remake bestBlack Beauty
The Browning Version
Lord of the Flies
The Stone Boy
Jacob Two-Two & The Hooded Fang
Heaven Can Wait/Here Comes Mr. Jordan
The Railway Children
Charlie and the Chocolate Factory/Willy Wonka & The Chocolate Factory
Finally, some stories I've seen more than 2 versions of; I put my favorite version in bold
Peter Pan (1924, 1953,
2003)
Oliver/Oliver Twist (1922, 1933,
1948, 1968, 1997)
The Secret Garden (1949, 1975, 1987,
1993)
The Dog of Flanders (1935,
1959, 1999)
Freaky Friday/Summer Switch (1976, 1983,
2003)
Little Lord Fauntleroy (
1936, 1980, 2003)
The Christmas Carol/Scrooge (
1938, 1951, 1970, 1984, 2004)
The Champ/The Clown/All About Ah-Long (
1931, 1952, 1979, 1989)
The Innocents/Turn of the Screw/The Nightcomers/Presence of Mind (
1961, 1972, 1992, 1999)
I think most people will prefer the first version of the films that they see as these films set the standard for what the story is all about. The remake has to work harder to leave a favorable impression. If a person didn't like the first film, they probably wouldn't see the remake anyway.