Incomplete and unsorted, here's a selection of the most engaging Harry Potter fanfiction I've come across. Assume a PG-13/R rating on average.
Crazy Little Things:
Hermione runs into Luna at university post-Hogwarts. One of my all-time favorites. Hermione x Luna.
A no-magic, modern AU where all of the Marauders except Remus attend a boarding school. Remus is sick and in the hospital. A wrong-number text from Sirius changes both their lives. Wolfstar.
Teddy Lupin is gender-fluid. It takes some time to come to terms with it. Teddy x Victoire.
Starts with Living With Danger (linked). Remus has a chance meeting at rock-bottom, and it saves everyone. This is a huge series spanning most of 20 years, 6 main stories, 16 in-sequence one-shots, and a bucket-full of AUs and crossovers.
Harry finds a horse in the Forbidden Forest. Harry x Luna.
Rewritten In Time:
After the war ends, Draco sends himself back to his 11-year-old body in an effort to fix things. Only, he's not the only one sent back. This series ties into Shattered and Over The Rainbow (linked from the main series where connected), because timelines tangle far too easily.
The Dream Trilogy:
The Multiverse has thin spots, and sometimes people fall between them. Eventual (one world's) Draco x (one world's) Hermione.
Children of the Dark:
A secretly-Dark Harry/secretly-Dark Hermione story with a satisfying reveal. Covers all 7 years; the link is to Year 1. Hermione x Draco.
Storm of Yesterday:
Harry and Hermione make a last stand at Godric's Hollow. Hermione x James.
Yule Ball Panic/Power Of Love/Heart And Soul:
Jasmine Potter needs a date for the Yule Ball, but can't ask Hermione for help this time. Because Hermione's the one she wants to ask to the Ball. The sequels deal with the aftermath in detail. Girl!Harry x Hermione.
Harry is raised by goblins. Harry x Hermione.
Harry gets detention with Umbridge and is dosed with veritaserum, and the truth comes out. But not just about him. Harry's reaction to most of this story is a bewildered "what the hap is fuckening?" And I've never seen this particular take on Umbridge anywhere else. Harry x Susan Bones x Hannah Abbot.
Behind Blue Eyes:
In the summer after 5th Year, Harry meets an old acquaintance from primary school. They run away together, escaping their awful home lives and finding solace in each other. But they can't hide forever. Harry x OC.
The Sum of Their Parts:
The war is over, but society hasn't changed. Harry realizes that real change takes drastic measures, and decides to take some of his own.
Don't Look Back In Anger:
Harry gets thrown in Azkaban after the Triwizard Tournament, and discovers A Power He Knows Not. Harry x Hermione x Luna.
Waking Fred Weasley:
Seamus throws a wake for Fred and the other casualties of the war. Harry x Hermione, Ginny x Seamus
What could drive them to destroy the wizarding world? Harry x Hermione (x Luna hinted at).
Night Out At The Kitty Kat Club:
Ginny drags Harriet with her to a lesbian bar to improve her prospects of joining the Holyhead Harpies. Girl!Harry x Ginny
The Weasley Twins get blitzed and turn Harry into a girl, leaving him asleep in a corridor outside the Slytherin common room. Draco finds him and isn't a jerk. (Girl!?)Harry x Draco.
Visions of Doom:
In the aftermath of his encounter with Moony in 5th year, Snape has visions of where his life is headed, and decides to make some changes. M/M (MPreg) but I don't want to give the ship away.
Snape is tired of being a pawn. A new prophecy provides possibilities.
Something goes wrong when the Weasley Twins try to get past Dumbledore's age line, and they find themselves in 1976.
Leo Inter Serpentes:
Harry gets sorted into Slytherin. Covers all 7 years; at this time, up to year 6 are completed. Harry x Draco.
Harry Potter and the Road Not Taken:
Harry Potter dies of old age and flatly refuses to go on the train to "the next great adventure" if it doesn't give him some damn peace and quiet. He ends up back in his 9-year-old body, only to find he's not the only one who's been sent back...
Harry and Draco are professors at Hogwarts. Then the world explodes, and the two of them are thrown into the heart of yet another world-saving adventure. I've never come across a take quite like this before, at least not done so well. Harry x Draco.