I’ve noticed these kinds of movies get real deep into a character. But they also display the external disparity between what’s said verbally to other characters, and what truths the protagonist tells the audience through narration.
“Leon Morin, Priest” is a good example.
A Christmas Story is the first one that comes to mind.
Are you talking Shawshank Redemtion? How 'throughout" must the narration be?
I imagine Goodfellas has to be the champ of that category.
“As far back as I can remember… I always wanted to be a gangster.”
Come to think of it, this is sort of a hallmark of Martin Scorsese in general. Taxi Driver, Gangs of New York, Casino and The Irishman all make use of this style.
When I saw the title of this thread I tried to come up with some titles before opening it. ACS didn’t cross my mind, which might indicate that it is so well done and integral to the story that it doesn’t feel like narration. It also might mean my memory sucks.
The best moment is in Casino, when Joe Pesci’s narration is interrupted by him unexpectedly getting hit with a bat (NSFW).
I haven’t seen that movie in a LONG time, but I’m not picky 
I just notice these kinds of movies being pretty good. I can’t think of an awful movie with narration…
“A Clockwork Orange” is another one. It was one of my very favorites, but I haven’t seen this in almost 20 years, but I feel awful when I re-review a movie after so many years, and then it not being as great as I remembered.
Days of Heaven was narrated by Linda, one of the trio of wanderers.
Saigon…
(do I really need to elaborate?)
Snatch and Sexy Beast were peppered with good narration bits.
For TV - best thing Ron Howard ever did was narrating Arrested Development.
What a great start to a movie…
“…Lookin’ fuh t’ings, searchin’ fuh t’ings, goin’ on a’venchahs…*”
Which is what I say when my wife asks me where I’m going… since it’s her favorite movie, how can she object?
…
*(adventures, if you’re wondering)
I loved the way she said “advenchas,” too.
Forrest Gump ?
“Fight Club”?
Tin Drum - by film’s end you wanted to smash little Oscar’s face in.
Limey - unique way of using narration by sliding in lots of dialogue over unfolding and concluding scenes.
Sunset Boulevard – Joe Gillis narrates his story
Little Big Man
These are all great films with varying amounts of narration:
La Jetée (1962, France, dir. Chris Marker)
(500) Days of Summer (2009, U.S., dir. Marc Webb)
Moonrise Kingdom (2012, U.S., dir. Wes Anderson)
Apocalypse Now
Fear and Loathing in Las Vegas
(heh - maybe I did) 
Not necessarily the protagonists, but I like how snippets of narration were provided throughout Broadway Danny Rose through Danny anecdotes from all the Borstch Belt comics hanging out in the restaurant.
William Hurt narrating fantasy scenes with the hot Sonia Braga in Kiss of The Spider Woman.
Kiss Kiss Bang Bang