The most obvious is Warhammer FRP 3E. The dice pool is still functionally a numerical system, but on multiple axis: Do you succeed at the fundamental task? (Axe vs Swords) Do you have beneficial or harmful side effects? (Eagles vs Skulls) Or wild results (Sigmar's Comets vs Chaos Stars)? Or Cost Extra (Fatigue drops & Extra Time Hourglasses)? There is a lot of room for narration, but also, many abilities have very mechanistic effects which the GM must then narrate how they occurred.
Similarly, One-Roll Engine (Reign, Monsters and Other Childish Things, and several others) games have a 2-axis dice pool system. I don't find it memorable. So I just looked it up, again. width is number of dice which rolled matching each other, and indicates speed; height, the number which matched up, is quality. And a success requires at least 2 dice to match which exceed the difficulty.
Another which does variable levels of success and failure is WWG's Storyteller system. The number of successes over the difficulty indicates quality of success. In some uses it is very mathematical; in others, very much up the the GM how much quality.
FATE and Fudge: both use the same ladder system, and encourage the GM to deal with the excess levels of success by narrating that.
To some degree, a great many games have some guideline at "better roll = describe better success" rather than the boolean Success/Fail, and a "Special Fail/fail/succeed/special succeed" is the most common. Many have suggestions (often in table-roll form) for special failures. A number add a marginal result.
My favorite, tho, is from Rogue Swords: Fumble, Fail, Marginal, success, overkill, critical. Fumbles hurt, and so do overkills; the difference is merely whether or not the action succeeded.