The Ringing Singing Tree has an initial quest, but it is not the main part of the story. A princess requests that a suitor find her the magical tree, the tree is given him with the condition that the princess love him, which she doesn't so he gets transformed into a bear. The princess is punished by being made ugly (hairs turns green) and then both meet in their transformed state and develop a relationship.
The story is not of the quest, per se, since the tree is found fairly early and without much trouble. 

A better case may be made of Magic Voyage of Sinbad (Sadko,) since it is your typical quest for the bird of happiness.

Ulysses should probably be disqualified, since he is not really on a quest for anything, unless it is to return home.

"There is a lot of money tied up in this film and people expect to hear a boom when something blows up, so I'll give them a boom."

George Lucas as quoted by Harlan Ellison's WATCHING
Last Edited By: hermanthegerm Sep 2 10 9:03 PM. Edited 3 times.