Unbroken Vigil

And when at last poor Emblus died
The osbick bird was at his side.
He was interred; the bird alone
Was left to sit upon his stone.
But after several months, one day
It changed its mind and flew away.

—Edward Gorey

If a character, often a hero, is in a coma, unconscious, ill beyond lucidity, has had their Soul stolen, has just had major surgery or whatever, then the odds are that they will have a friend or family member who refuses to leave their side. Ignoring all distractions and necessities, they will keep watch over them. Other well-meaning characters may urge the watcher to go take care of themselves, but their efforts always fail. As long as their charge is helpless, the watcher isn't going anywhere.

The watcher may keep their vigil until they pass out from exhaustion, usually slumped over their charge's bed. Sometimes they'll be reawakened only when their charge comes to. Other times they're found by a third party who was just coming to see if they could talk the watcher into going home and resting this time. You might well find this trope in a Sick Episode or Fever Dream Episode (though in the latter it can be tricky to tell what's going on...)

Occasionally emotion has nothing to do with it. Maybe the unconscious is the primary witness in a murder case, and the watcher is a detective waiting to spring the second there's a flickering of consciousness, for example.

Sometimes they are trying to save the unconscious person from the terror of Dying Alone.

Sister trope to Converse with the Unconscious, which often occurs in conjucture with the Unbroken Vigil.

Anime and Manga

Mikiya Kokutoh visits Shiki every damn week for two years that she spends in coma in Kara no Kyoukai. This even earns him the nickname "Puppy-kun" among the hospital staff.
Suzumiya Haruhi herself did this for Kyon after he gets stabbed but she thinks he fell down the stairs. It's complicated.
The titular character of Ranma ½ does this, in female form, for a sickly young boy. He had nearly died the night before because he would only accept his medicine if she delivered it with a kiss, and she felt responsible for the situation. Predictably, by the time he wakes up she has fallen asleep on his bedside, so the first thing he does is try to kiss her again, and gets a pounding for his efforts. He never did find out that it was the doctor who gave him the medicine.
In Magical Girl Lyrical Nanoha A's, Fate wakes up from having her Linker Core absorbed and discovers her familiar Arf asleep at her bedside; Lindy, who is still awake and has recently arrived, notes she has been there since the previous night.
In SHUFFLE!, when Asa wakes up in the hospital in Episode 20 after collapsing at the end of the previous episode, Rin is asleep at her bedside.
This happens in Player Kill, when Yonggi spends the night at the hospital after Gihoon gets blinded by Harata for sparing him.
Repeatedly done by Rei for Shinji in Neon Genesis Evangelion - an action that serves as a gauge of the closeness of their relationship throughout the series. Later on, when Asuka starts doing the same thing (Shinji does a lot of waking up in the hospital) the tension that results makes for some of the show's best scenes.
Whenever Naruto is in the hospital, Shikamaru is inexplicably sitting in his room. The first time, it was coincidental; he was there to visit Choji, who had been hospitalized for overeating. The second time, it was more likely intentional, as he presumably was also visiting the other injured members of his five-man team.
Yomiko in the Read or Die OVA does this with Nancy.
In Kanon, Yuiichi is staying by Makoto's bed, as she's supposed to die that night from high fever. He falls asleep, and wakes up in the morning thinking she died and he wasn't there for her - only for her to crack a smile at him.
In Soul Eater After Maka is paralyzed, she has to tell Soul to go home. The scene seems to imply that he hadn't left her before them. Similarly, after Soul had his torso bisected by Chrona Maka stopped short of doing this for him (and was implied to have stayed by his side while he was unconcious after surgery).

Comic Books

Happens in a Star Trek: The Next Generation Annual. Including the falling asleep by his side thing. Understandable in that not only is Data his closest friend, but Geordi's the one trying to repair him so it's probably better that he sticks around.

Fan Works

In the Code Geass continuation fic In the End, C.C watches Lelouch, waiting for him to wake up from his first death.
A lot of Ace Attorney fanfic has Mia acting like this to Diego, before eventually giving up.
In Kyon: Big Damn Hero, after Kyon had a bullet surgically extracted, Haruhi, Yuki, Mikuru and Tsuruya stay the night in the hospital keeping him company, even if Yuki had already healed him. As a bonus, both Haruhi and Yuki sleep in the bed with him.
Most fanfics that start after the third season of Castle ending have Castle doing this for Beckett.

Film

The Matrix Reloaded. Neo falls unconscious after destroying several Sentinels.

Maggie: He's in some kind of coma, but his vitals are stable. What about you?
Trinity: I'm fine.
Maggie: You could use some rest.
Trinity: No, I'm gonna stay with him.

Rocky II, when Adrianne falls comatose from stress over her husband's return to the ring and having to work while pregnant, Rocky begins a vigil broken only by visits to the church to pray, neglecting his training for the big rematch against Apollo.
In The Little Mermaid, after the final battle against Ursula, Eric lies motionless on the beach for a long time. Ariel sits on a rock and watches for him to wake up, finally causing her father to realize that she really does love the guy.

Literature

Happens multiple times in The Lord of the Rings. "He's hardly left your side these last three days."
Edward in Twilight doesn't leave Bella's side while she's in the hospital for several broken bones and loss of blood and maybe a few more things.
- Never the less, he's perfectly willing to leave her AFTERWARDS
Caitlin to Bran in A Song of Ice and Fire. She goes pretty mad with grief, but her madness, and his dire wolf saves his life from the hired idiot of a killer Joffrey sent.
In Warrior Cats, Cloudtail does this with Brightpaw in A Dangerous Path when she gets attacked by dogs. Eventually the medicine cat has to order him to eat and rest.

Live-Action TV

In the Bones season 3 finale, the team's commitment to their intent to constantly watch over their injured co-worker Zack leads them to assigning shifts. In doing this, they unwittingly prevent him from escaping, as he was working with the Serial Killer and had engineered the explosion that injured him.
Done in an episode during the first season of Titus, Erin refuses to leave Christopher's side after his dragster crashed, leaving him braindead. Played for laughs in that her refusal to even let go of his hand leads her to almost drag him completely out of bed.
After he's injured by a gunshot wound, Melinda stays by her husband Jim's bedside for hours while he's recovering from surgery, to the point where she falls asleep. When she awakens, Jim lovingly says hello unfortunately it's because he's trying to explain as gently as possible that he's died of an aneurysm.
Played straight and subverted in Hannah Montana where Billy Ray promises to be awake and smiling when his daughter wakes up only to predictably fall asleep. The subversion comes when Jackson tells him that there's pie in the cafeteria. Not only does this wake him up instantly, but he runs off instantly and doesn't come back until after she's left the hospital.
Series 5 of Doctor Who has what might be the best example: Auton!Rory spends almost two thousand years protecting the Pandorica that holds Amy, becoming the legendary Lone Centurion in the process.
Any given episode of The X-Files has Mulder or Scully doing this.
- This actually has a bit of Fridge Logic to it. During the entire series, we see Mulder and Scully sitting by each other's respective bedsides without any problems from staff. But during season 8's "Empedocles", Mulder gets kicked out of Scully's hospital room for not being immediate family (specifically, he's not her husband)--repeatedly. Which begs the question...how were they able to do it before?
  - Given the way those two act, nurses and doctors likely just assumed they were married. The nurse in "Empodocles" was probably just the first nurse who bothered to ask!
When Donna Moss is critically injured in a bombing in Gaza, Josh Lyman drops everything (keep in mind he is the Deputy Chief of Staff to the President of the United States) to go to Germany and be with her. He stumbles into the hospital exhausted from the flight and refuses to leave her bedside until she wakes. It takes these two a further season and a half to get together.
Not a completely unbroken vigil, but an episode of Mysterious Ways has a man continue to visit his comatose wife every day for sixteen years.
In the Torchwood season 1 finale, Gwen will do a vigil when Jack is assumed dead.

Radio

Ianto to Jack in the Torchwood radio play "The Dead Line".

Video Games

Persona 3 liked this one: first, it was Yukari watching over the Main Character during his week-long coma after he first summoned his Persona, saving her life in the process. Later in the game, Junpei would utterly refuse to leave Chidori's side throughout her hospitalization/confinement, despite her own insistence. In fact, it's because of this dedication that Chidori warms up to him, then realizes that her life is worth something to someone, which goes against everything she believed in while she was a part of Strega.
- Aigis herself takes this to a rather creepy extreme, watching over the Main Character all the time, to the point of timing herself on how fast she can pick the lock to his room each night.
Squall Leonheart in Final Fantasy VIII did this for Rinoa, when possession by Ultimecia left her in a magically-induced coma. Frustrated that he was getting nowhere watching over her at Balamb Garden, though, he took the vigil on the move by carrying her across the world to Esthar.
- In Final Fantasy VII, Tifa did this for Cloud.

Web Comics

In General Protection Fault, Fooker remains awake at Maddie's side for several hours until she comes to from being poisoned on a mission to capture Dr. Nefarious, and eventually falls asleep to catch up on lost sleep.
In It's Walky!, Joyce is shown to have brought a bedroll and food to stay near the Martian Resurrection Machine that is reconstructing Walky.

Western Animation

One Futurama episode has Leela fall into a coma, and Fry stays by her side for weeks talking to her. When she wakes up, they hug, and then:

Leela: (whispers) You could really use a shower.
Fry: (whispers) So could you.

- There was also Fry's puppy, who grew old and fossillised waiting for his master to come back.
In ReBoot, Enzo keeps a vigil over AndrAIa after she falls into a coma from having part of her code stolen by a Web Creature.

Real Life

Happens in real life far more often than many people realise if the stories are to be believed.