Each animation is a THREE.AnimationClip instance. JavaScript’s JSON.stringify() is really only useful for primitive objects with keys and values, not for instances with methods and behavior and inheritance. Instead, use:
var jsonObject = animations[ 0 ].toJSON(); // creates a primitive object with basic animation data
var jsonString = JSON.stringify( jsonObject ); // serializes the object as a string
To recreate the AnimationClip later, you’ll need to use var clip = AnimationClip.parse( jsonObject ).