Some characters have True Companions. Some characters have social circles (however limited they may be). Alice doesn't have that. Alice has Bob. Just Bob.
There are two possible variations on the Alice character here:
Alice is shy or a bit of a misfit, but ultimately a nice person. Bob is the only one to take the time to get to know her and realize that she's actually pretty cool.
Alice is a Jerkass or otherwise hard to get along with. Bob is the only one willing to put up with her shenanigans.
There are also two variations on the Bob character:
Bob is a fellow misfit who was drawn to Alice because neither of them had any other friends.
Bob is the one with all the social skills, and may try to help Alice make more friends and/or smooth over situations where her behavior has caused problems.
While Alice-1s often go along with Bob-1s and Alice-2s with Bob-2s, they can be mixed and matched: a shy Alice might have a social Bob who tries to draw her out of her shell, while a mean Alice might have a shy Bob who's stuck with her because he's lonely or doesn't have the nerve to ditch her.
In many stories, an Only Friend doesn't stay that way for long: a nice Alice will usually have learned to make friends by the end of the story, while a mean Alice will often find Bob finally getting fed up and leaving her.
Compare with Living Emotional Crutch, which might be at play in an extreme form of this trope.
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Anime and Manga
Jerkass Haruhi Suzumiya notes with pride in The Vanishing of Nagato Yuki-chan that only Kyon speaks to her informally. In the Haruhi Suzumiya series proper, Kyon's sister (that's all we know of her name) and Tsuruya are the only persons who are friendly towards Haruhi, without being burdened with Haruhi's secret.
Shy Yuki Nagato (in either series) doesn't feel comfortable going to the library without Kyon.
In Beelzebub Furuichi is Oga's one and only friend, due to the latter's inability to stop fighting and rough personality.
Himekawa's only friend was Kugayama, until he betrayed him, which really screwed him up and left him unable to heal until he met Oga. That said friend was also in reality a chick and in love with him did not help.
It gets a little darker than that (naturally). She was friends with all of the girls before but with each new timeline she got closer to the edge of insanity and stopped caring about them altogether and admits she doesn't "give a damn" if some of them die along the way to save Madoka. It could be assumed that the reason she never stopped caring about Madoka is because in every timeline Madoka is literally the only one who never turned on her and was always nice to her no matter the situation they were in, unlike the other girls who would call her a liar and sometimes attempt to kill her in a bout of depression. Sayaka in particular could be very cruel to her.
Homura: "Madoka, my only friend… if it's for your sake I don't mind being trapped in this endless maze forever."
This seems to be a recurring element of the series. It's shown in The Different Story that Mami and Kyouko once had this relationship, and them breaking it off was a major factor in Mami's possessive attitude at the time of the series, and why Kyouko was so desperate to get close to Sayaka. In Rebellion, it's revealed that Mami's greatest wish was to have a friend, hence the presence of Bebe/Charlotte/Nagisa, who acted as this toward her.
In Legend of Galactic Heroes, it can be assumed that this trope applies to Reinhard and Reuenthal, with Kircheis and Mittermeier respectively. The latter two are sociable enough to at least be able to have friends aside from them, but Reinhard is far too busy realizing his ambition to befriend anyone but Kircheis, and Reuenthal's trainwreck of a psyche doesn't let him get close to anyone aside of Mittermeier who becomes his Living Emotional Crutch.
Reinhard does open up a bit to others as the story goes on, most notably Hilda, Mittermeier and Emil.
Applies to Alice in ARIA to a degree. Since she's extremely unsociable and cold towards anyone, it's really only Akari and Aika that she can truly call her friends. She insists that they are different from all the other girls, although this is mainly because of her warped perception of the people around her - She thinks she's disliked and envied for being talented, when in reality most people admire her.
In Mai-HiME, at the start of the series, Shizuru is Natsuki's only friend.
In both the anime and Light Novel of Fate/Zero, Gilgamesh cites this as the reason he refuses Rider's offer for them to join forces. Enkidu is the only friend he has ever had, and the only one he ever wants to have.
Koyuki was Ginta's only friend at the beginning of MÄR. Weirdly enough, it's implied that she's just a friendly classmate in the opening, but as we see more and more flashbacks, it turns out she's been his only friend for a long time.
Dragon Ball Z - after saving Gohan's life by jumping in the way of an energy blast, the dying Piccolo admits that the kid he abducted and put through Training from Hell is the only friend he's ever had.
From the backstories of the girls in K-On!, it seems like before they met each other and became True Companions, nearly all of them had only one friend to depend on. Mugi (Alice-1) and Sumire (Bob-1) literally had no one else except each other growing up, because both of them were home-schooled and heavily sheltered. Ditzy Yui (Alice-1) doesn't seem to have had any other friends before high school except Nodoka (Bob-2), and Mio (Alice-1) was so shy and quiet as a child that only Ritsu (Bob-2) ever managed to break through her shell.
In Code Geass season 2 Lelouch has lost his love interest, his sister, his best friend, and has a confusing relationship with the rest of his friends. The only person who truly understands and trusts him is CC. Then she loses all her memories and personality. Because of this Lelouch takes over the pre-episode speech for the next few episodes, and the bitterness and anger in his voice just tells all.
Helena Bertinelli, aka the Huntress, had only one friend, the Black Canary. This was primarily because Huntress was a borderline-sociopathic misanthropic loner, so she was an Alice-II. Eventually, Huntress seemed to be forming friendships with a couple of other characters, notably Renee Montoya (the new Question), and Oracle. All this, of course, was in the old DCU; now we're back to Helena Wayne.
Transformers: More than Meets the Eye: Brainstorm is a quirky Mad Scientist with a Lack of Empathy, a penchant for making weapons so utterly unethical he's built up a reputation for them, and a Heroic Comedic Sociopath streak. Chromedome is his only friend, a fact he acknowledges to have Brainstorm do him a few favors. Brainstorm actually cares for Chromedome, and tries to talk him out of erasing all the memories of his now dead lover Rewind to spare himself the pain. He admits to seeing him do it 3 times before, and is concerned about him, worrying about his mental health and his lack of coping abilities. He gives Chromedome Rewind's last message, and leaves him to do what Chromedome thinks is best.
In Watchmen, Rorschach's only friend is Nite-Owl, and even his patience is really tested with Rorschach's personal habits and extreme opinions.
Jenny to Forrest Gump in their youth. And Bubba to him in the war. And then Lt. Dan for a time after the war.
Back to the Future: Marty McFly is all Doc has, except for his dogs. Doc is an Alice-1, a decent and loyal man who happens to have the reputation of being a Mad Scientist. Marty himself is implied to be part Bob-1 and Bob-2, as he is a Cool Loser who nonetheless knows enough about making friends to have the fellow members of his garage band.
Dante to Randal in Clerks and Clerks II, which is why Randal is so rattled by the thought of Dante leaving in the second one.
Randal: You think I wanna start making friends at my age? Christ, who would want me as their friend? I hate everyone, and everything seems stupid to me, but you were always the counter-balance to that. The guy who was the yin to my yang. Now what the fuck am I gonna do for the rest of my life?
In the first Harry Potter book, Ron is Harry's only friend. Harry probably isn't Ron's only friend but since it's Harry's story we don't see Ron outside his interactions with Harry (though Harry probably is Ron's best/most important friend, since while we don't see the story from Ron's point of view, we do know that Harry spends most of his time with Ron, and hence the logical inference is that Ron also spends most of his time with Harry). Hermione begins the year friendless and isn't friends with Harry and Ron until midway through the book.
It's hinted that Ron's younger sister Ginny was this to Luna Lovegood, as one of the few people who didn't straightforwardly reject her for being odd.
From what we see of Snape's memories, it's made clear that his only real friend was Harry's mother, Lily, until she ended the friendship at the end of their fifth year because he called her racist epithets when she was trying to help him.. In his adult life, if he has a friend, it's Dumbledore; he's certainly the only person who truly knows and likes him.
Sherlock Holmes' idiosyncrasies and general lack of interest in other humans except as puzzles ensures that Watson is his entire social circle.
Watson: That was surely the bell. Who could come to-night? Some friend of yours, perhaps?
Holmes: Except yourself I have none. I do not encourage visitors.
By "The Final Problem," Holmes appears to have some Character Development, writing in his note to Watson how his coming death will cause grief to Holmes' friends, and in "The Adventure of the Six Napoleons" enjoys the regular visits of Inspector Lestrade.
John Cleaver from the I Am Not a Serial Killer trilogy is friends with another social outcast Max in order to try and avert falling into Loners Are Freaks. Despite his sociopathic nature, John makes other friends in thesequels but is so focused on his quest he pretty much still only has one friend at a time, leaving poor Max friendless.
In The Princess Bride, Fezzik and Inigo are each other's only friend (at least until they team up with Westley and Buttercup). However, as these things go they both seem fairly easy to get along with, so it probably has more to do with that they're busy doing Vizzini's bidding, Inigo is obsessed with avenging his father, Fezzik is used to being an outcast, and it emphasizes their Heterosexual Life-Partners relationship.
Francis is Michael's only friend in The Traitor Game. Not the other way round, though; there are mentions of Francis's other friends. They're both a mix of the two types. Michael don't easily make friends (and when he does form a friendship, he manages to break it), but that's only because he was bullied at his previous school. And while Francis is a Nice Guy and a Cool Loser, he's also a bit of a misfit - possibly because of his friendship with Michael and his homosexuality - and he too sees Evgard as an escape from everything else.
Before the main events of The Hunger Games, Katniss's only friend was Gale, unless you counted the sort-of-friendship she had with the mayor's daughter, Madge, as they both were loners at school and therefore got paired up together in activities by default.
Early on "In Plain Sight," Marshall and Mary proclaim each other to be this. Ultimately, it seems like their circles have expanded... by maybe one or two people each.
In Please Like Me, Josh's boyfriend Geoffrey admits that Josh is his best and only friend. Which Josh finds odd.
Thomas Lynley and Barbara Havers. He is her only true friend. Funny enough it also works in reverse - while Lynley does have one or two other close friends, he is nearly as difficult to work with as Havers is. They ultimately wind up as True Companions, via complementing each other surprisingly well, presumed betrayal, a near drowning, and more tragedy than any two people should ever have to go through.
In Star Trek: The Next Generation Q claims that in all the universe Picard is the closet thing to a friend he has. Apparently the Q Continuum from which Q comes from has become far too rigid, boring, and predictable for his tastes, and Picard is one of the few interesting things going on in the universe.
Merlin: Both Gwaine and Arthur have said that Merlin is their only friend.
Scully (Bob-2) and Mulder (Alice-1) of The X-Files. Scully seems to have peripheral friendships that come up when convenient, but for the most part they are each others' only consistent, trusted friend.
In FoxTrot, Marcus is basically Jason's only real friend. The only other peer he ever spends any time hanging out alone with is Eileen (on a few rare occasions), and his feelings towards her are so conflicted - partly because she's also an academic rival and partly because he's still convinced Girls Have Cooties - that it's hard to call it a truly mutual friendship.
A Running Gag in the one man play "Shylock" is that Tubal is Shylock's best friend - well, his only friend.
In Chess, Florence is Freddie's only friend, although by the end of the first act even she has gotten fed up with his Jerkass behavior and left him. (Which, given what we see of their interactions, is less surprising than the fact that she was with him in the first place.)
Kingdom Hearts II gives us the scene of Axel's death, where he tells Sora that Roxas was the only one he liked, and basically his only friend, though that wasn't really the case, since Xion was also his friend along with Roxas, though when she "died", all memories of her disappeared.
In Ghost Trick, Sissel was Yomiel's only friend in the original version of the previous 10 years. This eventually changes.
In No More Heroes, Bishop Shidux is described as Travis Touchdown's only friend. His murder by the the Big Bad convinces Travis to reenter the assassin ranks in the sequel.
In Rockstar's Bully, main character Jimmy Hopkins is shy guy Petey's only friend. Presumably this changes after the end of the game when Jimmy convinces Dr. Crabblesnitch to make Pete the new head student at Bullworth.
If Shepard is female and begins a romantic relationship with Garrus in Mass Effect 2, Garrus tells her she's his only friend in the galaxy. Thane admits to Shepard that s/he's the first friend he's made in twenty years. Liara suggests this when she asks Shepard for patience on Illium, saying she doesn't have enough friends left to lose him/her.
Due to his attitude problems, Barry Wheeler is Alan Wake's only friend, if you don't count his wife. The point is brought up several times throughout the series, with the imaginary Barry from The Writer, mentioning how pathetic it is that Wake's only real friend is his agent, and Alan admitting, through a manuscript page in American Nightmare, that, while he doesn't make friends easily, he trusts Barry completely.
In Dragon Age II, Fenris can tell Hawke they're the only friend he's ever had before the final battle. In practice, he's on fairly friendly terms with most of the party's non-mages (and plays cards with Aveline's husband every week).
Elisabeth Blanctorche and Ash Crimson have this kind of relationship in The King of Fighters. So much that Ash's big ass Batman Gambit (which included de-powering Chizuru and Iori, attacking Kyo, joining Those of the Past, antagonizing Elisabeth, manipulating everyone and ultimately getting himself ret goned was to keep Elisabeth safe.
In the sequel, Super Dangan Ronpa 2, Sonia Nevermind is this for Gundam Tanaka. Gundam rarely socializes with the other students when there isn't a problem, claiming that his only friends are his pets and that he doesn't need human companions. Not including the optional friendship he can have with Hinata, the only student he eventually befriends and shows respect to is SHSL Princess Sonia Nevermind, who shares his interest in animals and occult things. She's also the only person who Gundam doesn't refer to with any sort of contempt or condescending attitude, and the only one he willingly hangs around and communicates with.
In Fate/EXTRA, this is Gilgamesh's stance on Enkidu. Since Enkidu was originally a weapon whose life was "without worth", Gilgamesh proclaimed that Enkidu would be his only friend for all eternity, thus granting him an absolute and undiminishing value.
Kratos from God of War appears to only have one friend in the entire series who doesn't either die too quickly for him to develop any kind of bond with, betray him or get betrayed by him in Orkos from Ascension.
Well, Orkos and Pandora. Pandora has enough of an impact that he finally starts to take responsibility for his crimes and feel remorse.
In Katawa Shoujo, Lilly is initially Hanako's only friend, until Hanako falls in love with Hisao in her own route, and befriends Hisao and the members of the Newspaper Club in Lilly's route.
Shizune and Misha are also this to one another unless you take Shizune's route, although Hisao says in Lilly's route that he considers both of them his friends.
In Bionic Heart, the only friend Luke seems to have at work is Tom, who is the social type trying to help Luke get out more. This makes it all the worse when Luke learns that Tom was having an affair with Helen, his girlfriend. And then if Luke chooses to tell him about Tanya, the fugitive android hiding in Luke's apartment that was manufactured by their company, Tom sells Luke out to the company to ensure his own safety.
In Homestuck, Karkat is Eridan's only friend (and even that's stretching the definition a bit) after Feferi breaks up with him. And even Karkat abandons him after he murders Feferi and Kanaya and hopesplodes the Matriorb.
Kanaya is Vriska's only friend after the Incident that led to the split-up of Team Scourge, for reasons like this:
AGnote Vriska: Lousy st8pid godd8mn supportive fri8nd!note This is right after Kanaya tries to help Vriska, who isn't being sarcastic here and is genuinely pissed off about it.
Bender: Fry, of all the friends I've had... you're the first.
Dan Vs.: Chris is Dan's only friend. Ted is occasionally mentioned, but has yet to actually appear on the show, probably because Dan is usually mad at him.
The first season finale of Megas XLR reveals that Non-Action Guy Jamie resents being taken on dangerous missions, and the main reason why he doesn't simply leave is because Coop's his only friend.
Daria: Daria and Jane (a rare combination of Alice-2 and Bob-1, respectively) are this to each other in the earlier seasons of the series. They eventually form other friendships, especially after Tom enters the mix...
Well, Jane does - Daria only really has Tom and (sometimes) Jodie.
Teenage Mutant Ninja Turtles (2012): Although Leatherhead calls the four turtles his friends, only Michelangelo (the one who actually befriended him) reciprocates this. The others are at best trusting or at worst afraid (Donatello being the latter).