I could make a very long list here, but will limit myself to three
1. An old favourite, which is a wonderful fleshing out of my favourite moment in "Lord of the Rings."
The Falcon and the Star - Raksha the Demon
http://www.tolkienfanfiction.com/Story_Read_Head.php?STid=695
Description:
It is March 15, 3019. After the longest week in his life, Aragorn is called by Gandalf on an errand of mercy. One weary warrior meets another in the Houses of Healing and other, farther places; and their lives will never be the same. Most of the story (but not all) from Aragorn's POV. Some dialogue unavoidably quoted and paraphrased from Return of the King. Bookverse.
2.A recent story, which deserves more readers.
The Oliphant's Tale by Lady Branwyn
http://www.fanfiction.net/s/7663089/1/The_Oliphaunts_Tale
What happened to the oliphaunts who fought in the Battle of the Pelennor Fields?
3. Another Old favourite
Blossoms by Girl in the Red Jacket.
http://www.fanfiction.net/s/1897638/1/