What a great question! From an early historical standpoint, I can think of several cases where this has happened. I'd be interested in more answers, and especially later historical examples.
In the Middle Ages and Renaissance, accidentals were often not notated, with the composer relying on the performer's knowledge of musica ficta to provide the correct pitches. Ornaments were also not typically notated, and were left to the performer's taste; there are treatises from the time (e.g. Silvestro Ganassi's Opera intitulata Fontegara) on how to tastefully improvise ornaments and other elaborations (called divisions).
Starting in the Baroque era, there was an increasing trend to write out at least partly ornamented lines (though much was still left to taste). However, articulations and dynamics, while occasionally indicated when important, were often left unmarked. On the other hand, there was also the creation of basso continuo, which took a relatively non-prescriptive, improvisatory approach to accompaniment.
Beethoven was one of the first composers to specify precise tempos in terms of metronome beats (e.g. =120), rather than a general feeling (e.g. Allegro, Andante...).
Edit: I think Jazz probably provides a good modern counter-example, where there is an incredibly non-prescriptive notation. You might be given a lead sheet with chord symbols, and expected to decorate the melody line, substitute chords, and improvise various fills and counter-melodies on the fly.