The Doctor and River Song. Two time travelers, traveling in opposite directions. Every time they meet, one of them knows the other one more, while the other one knows the first less. And in the end, River's end, the Doctor can't even be there to tell her that he still loves her.
The Doctor's first meeting with River (that he remembers) is in his tenth incarnation, in The Library, where River dies. When meeting him, River acts as if she knows the Doctor and it is hinted numerous times in this episode that she isn't just a friend, but something more. At the end of the episode, River sacrifices herself to save The Library and before doing so, she tells the Doctor that he'll meet her again in his future and realizes the heartbreaking truth that all this time that she's spent with the Doctor, he always knew how she'll die.
After regenerating and traveling with Amy Pond for some time, the Doctor encounters River Song in the Byzantium. He acts very cold to her through the episode, probably because of not knowing who River is and because of knowing that she's a part of his future and the Laws of Time have some rules about that. During her time in the Byzantium, River attempts to flirt with the Doctor, though he does not take it well. Only in the ending scene, River, not directly, calls the Doctor a "very good man. The wisest man I've ever known,"
Under Stonehenge, where the Pandorica opens, the Doctor and Amy meet River Song once again, also being joined by a plastic version of Rory Williams, Amy's boyfriend who she forgot as a result of him being sucked into a crack in the universe in a previous episode. River is seen wearing Cleopatra's clothes, possibly indicating that she had sex with her. Who knows. River again flirts with the Doctor, this time, he sometimes takes this well and flirts back. River makes sassy comments about the Doctor's fez, holds hands with him and makes a Dalek beg for mercy. The Doctor restarts the universe by committing suicide, but Amy remember's him through the Journal so everything's alright.
Continuing their tangled timelines, River and the Doctor, along with Amy and Rory, meet at Lake Silencio, Utah, America. This time, the Doctor flirts shamelessly with River, both of them enjoying each other's company. During the story, the Doctor uses the TARDIS to catch River from falling to her death from a building. At the end of the story, the Doctor returns River back to her cell in Stormcage, a very secure prison that River escapes out of a lot, and before the Doctor leaves, River kisses him passionately on the lips.
After wondering who River Song is for so, so long, the Doctor, a good man, goes to war. At Demon's Run a child is born, being stolen from her mother after a terrifying battle. They train her, into a perfect psychopath, but she falls in love with the one she's supposed to kill.
Furthermore, in canon works outside of the TV series, the tenth regeneration of the Doctor is seen fully clothed in a bath tub with River, for unknown reasons. Afterwards he kisses her on the cheek. In other instances the Doctor is seen thinking of River, mentioning her as his wife and once it was heavily implied that River sent the Doctor nudes.
On AO3 it is the most written relationship for both characters. It is also the second most written ship in the Doctor Who (2005) tag.