In the Dragon Ball series, Vegeta always wants to train alone, even though he could potentially become stronger if he trained with a sparring buddy. Perhaps justified given that the only potential sparring partner who'd be able to keep up is the guy he desperately wants to surpass; Vegeta wants training that he alone will benefit from.
In Switch Hal does this and, despite emphasizing teamwork, rookie Hiki also tended to do this.
Kouji Minamoto in Digimon Frontier for the first five episodes. And then he gets his ass handed to him by Grumblemon in episode 6 but he doesn't accept being part of a team until episode 7 after many fights with The Hero.
Jyu Viole Grace from Tower of God rejects the option to recruit 7 team members for a team based test and asks the administrator to fail everybody else. This is because he is working for his boss because his friends are being held hostage and he really doesn't want to bring more people into this. Ultimately, he becomes an ally of Ja Wangnan and the rest because of his good nature. And that's good, otherwise Ja would have been eviscerated and his organs would have beeen sold off.
Phoenix Ikki from Saint Seiya is a textbook case of this. Years after the manga debuted he's still off doing his own thing away.
This is Chang Wufei's attitude throughout Gundam Wing, until an encounter with the ZERO System convinces him that teaming up with the other pilots is the best course of action. He goes right back to being a loner in the post-series manga, culminating in Endless Waltzwhere he does a Face-Heel Turn because he refuses to change his way of thinking.
A common theme in sports manga/anime is having a very talented player say this repeteadly and refuse to team-work, often to see him/herself forced to when things go hard. Examples are: Kaede Rukawa from Slam Dunk; Koujiro Hyuuga, Louis Napoleon and Natourezza from Captain Tsubasa; Ryoma Echizen and Kaoru Kaidoh (at the beginning) from The Prince of Tennis.
Played for Laughs in the case of Takeshi Momoshiro and Ryoma, who try to play doubles and simply cannot do so in the Fudomine arc. Momoshiro gets over that later and becomes a decent doubles player, but Ryoma stays a singles player.
In Pandora Hearts Sharon calls Break "Mr. One-Man-Show" because of his tendency to refuse help and go off on his own. He even says himself that he doesn't know how to fight as a team. He's actually pretty good at taking care of business by himself, but due to his recent blindness and deteriorating overall physical condition, his willingness to rely on others, however slightly, is part of his character development.
Blood+: After the Time Skip, Saya, as part of her Heroic BSOD, makes this a priority and goes out of her way to avoid working with her pals in the Red Shield because, after losing Riku, she doesn't want to feel the pain of loss anymore. Eventually, she snaps out of it.
This was actually the Fatal Flaw of Heartcatch Precure's Yuri Tsukikage. She was so confident in her abilities as Cure Moonlight, that she ignored her partner, Cologne, in finding allies. End result? A massive Curb-Stomp Battle leading to Cologne's death and her temporary depowerment.
Barnaby holds this stance for the first third or so of Tiger & Bunny, insisting that Kotetsu's only good for getting in the way and being annoying. He changes his tune completely post-defrosting, to the point that he doesn't see any reason to be a Hero unless Kotetsu's there with him.
Yu-Gi-Oh!!: Seto Kaiba often utters this when someone offers their help. He actually despises the idea of team work because he hates relying on others, especially his rivals.
This is Batman's favourite phrase, despite him nearly always having at least one sidekick, several other members of the Bat family, and regularly working with the Justice League.
For all the Broken Base over the Hush arc, it did a very good job of pointing out that for a supposed loner Batman has a huge surrogate family (from Alfred to the Robins to the JLA to random people from the Silver Age)
And now he's formed Batman Inc., following a moment of self-realisation:
"The first truth of Batman. It had to be one I don't like to admit. The gunshots left me alone. For years I was alone in the echoing dark of that wait. But something else defined the exact moment Batman was born. The first truth of Batman. The saving grace. I was never alone. I had help."
In the former page quote, Spider-Man and Daredevil are open to the occasional team-up but do work pretty exclusively alone, and attempts to change that generally don't work out. But Wolverine was created for a team book, and therefore almost always works in groups. He doesn't seem to have noticed. In fact, he is currently a member of four different teams.
Though these days, Spider-Man is a regular with The Avengers, having proved his worth as a member of Earth's Mightiest Heroes.
In Prisoner of the Horned Helmet, the main character had his entire tribe destroyed when he was 9 or 10 (he doesn't know which). as a result, he swears to never depend on other people, and will not work with anything that has two legs. He does have a pet wolf, though.
In Andre Norton's Catseye, Troy did not make friends easily among the other refugees and is isolated.
Lisbeth Salander in "The Millennium Trilogy". From the beginning of the first book, it is known that she works as a private investigator, and never comes into the office unless to deliver a report. She talks to people only when necessary and cuts off all contact with Blomkvist in the second book because she has fallen in love with him, then loses contact with the few friends she has when she travels around the world in the second book. When she comes home after more than a year she remembers that she had a casual girlfriend, Miriam Wu, who she didn't say goodbye to.
When Blomkvist repeatedly offers her help in the second book when Lisbeth becomes the Prime Suspect of a murder investigation Lisbeth denies him, because she looks out for herself, never trusting anyone to help her. This leads to her taking on Zalachenko and Niedermann by herself, and getting shot in the head. The reason she survives, is because Blomkvist finds her in time to call an ambulance, and duct tapes the wounds on her body, to which she responds "Kalle Fucking Blomkvist"
Live Action TV
The Golden Girls: Sophia says this a few times when someone interjects as she's telling a Sicily story.
After having lost two companions in tragic ways, the Doctor insisted on traveling alone, claiming that he didn't want to have his hearts broken again.
Then subverted with End of Time.
Wilf: Have you got anyone?
The Doctor: No. Traveling alone. I thought it was better, I thought... but I did some things, they went wrong... I need... [voice breaks]
Abby of NCIS. Her bosses occasionally forced her to take on lab assistants, but she drove them away. After one of them tried to stab her, management decided to leave well enough alone.
Burn Notice: Jesse has this problem. He used to be out in the field but got put behind a desk mainly because he didn't work well with others.
Dexter: Dexter tells Miguel this in season three and Lumen in season five, but it didn't really work out either time.
On The X-Files, Mulder is originally portrayed this way. He hates the fact that he's been partnered with Scully and spends most of the first season alternating between annoying the hell out of her and being nice to her. By season two, though, he doesn't work alone...he just works with Scully. And if he can't work with Scully, he wants to work alone. He doesn't like be partnered with anyone else even for a short time. Scully does the same thing in season 8 when partnered with John Doggett.
Andros from Power Rangers in Space is portrayed this way, especially when he's introduced. He's not thrilled to run into the former Turbo Rangers, and fully intends to send them back to Earth and out of his way. Part of his Character Development is learning the value of working with a team. He does have Freudian Excuse though - his former partner, Zhane (the Silver Ranger), was severely injured in a fight a few years ago and is now in a coma.
Merrick does this as well; he shows up to fight, but the rest of the time he stays as far away from the team as possible. Gradually eases over the series, and by the last few episodes Cole and Alyssa are turning up at his work and he agrees to join a party on the Animarium.
Robo Knight is another one that needs to warm up to interacting with others.
Sailor Mars pulls this after she receives her powers in Pretty Guardian Sailor Moon, though she kept getting pulled into battles with the other scouts, and Usagi wasn't about to stop trying to be her friend.
Sailor V/Venus pulls this too, when she's pretending to be the Moon Princess the others are supposed to protect.
Gorgeous George was in several tag matches over his career, but he wasn't very successful in them.
While Survival Tobita, the star of Saitama Pro Wrestling Company, has been in tag matches in other promotions, he's known primarily as a singles wrestler.
Destroy The Godmodder uses another variant. In the first game, several different players attempted to assist the godmodder. Each time it resulted in the godmodder stating that he did not want anyone else doing his job, and then pounding them into the ground, or throwing them into the sun.
Kopaka says this, word-for-word to Pohatu in BIONICLE.
Urdnot Wrex points out that he's never been much of one for working in an organized army, as "things get... messy." He works best alone, or "in very small groups." Fortunately, Shepard's team counts as such.
Previous to that, the turian Spectre Nihlus declines to accompany Commander Shepard's ground team on the mission in Eden Prime, claiming that he moves faster on his own. This ends up getting him shot in the back of the head by his colleague Saren.
In Dissidia: Final Fantasy, Squall works alone. The game manages to look at his attitude from a couple of different angles; although the early parts of Squall's story mode seem to set it up as a problem he needs to get over, and he eventually ends up working with Bartz and Zidane, the latter half of his story mode reveals that his choice to travel alone isn't because he doesn't trust the other heroes, but because he prefers to help them by taking on the villains by himself and eliminating at least some of them before they can cause the others problems.
More complicated than that, even. He fights alone because he doesn't want the burden of directly protecting people (it's too stressful, and far too easy to make a mistake and get someone killed), and because he trusts his friends to be able to take care of themselves. He knows they're strong and capable, so he doesn't worry about them unless he has to (such as when he saves Zidane and Bartz). He's not opposed to helping others, and in fact he does it without hesitation (running to rescue Bartz with Zidane, for example), but he just finds it's easier to push his limits when he's on his own.
To elaborate, the WarriorofLight confronts Squall half-way through his story and calls him out on his apparent lone wolf attitude, mistaking it as a sign of distrust. After their battle, the Warrior of Light realizes that Squall's intentions aren't as selfish as they appear to be, with the latter providing the above-mentioned explanation for his actions.
Thief II: The Metal Age: Garrett says it near the end; there's an Aesop present, and it probably sticks, though he backslides furiously throughout most of the sequel.
In both Valkyria Chroniclesgames, anyone with the "Lone Wolf" trait loses combat effectiveness when in the presence of other allies. Exemplified by Nils Daerden from the first game, who also has no other character to list as a "friend".
Herschel Biggs in L.A. Noire before he eventually accepts Cole Phelps.
Epicly subverted with a big dose of lecture in the beginning of Wild ARMs 3. After the intro sequence and all four character were in a bar, Jet announced that he works alone. Virginia would have none of it, replying directly with "What have you achieved while you were alone?" Make no mistake, it takes a significant chunk of the game for Jet to truly have a meaningful character development, but at least the intent is there.
"Let me tell ya a story. Once upon a time, I had somebody that I cared about. It was a partner. Somebody I had to look after, and in this world that sorta shit is only good for one thing: gettin' ya killed! So ya know what I did? I wizened the fuck up! And I realized it's gotta be just me!"
Butterfly: Parodied; Bat-Knight, a Captain Ersatz of Batman, says this constantly despite being surrounded by sidekicks and etc.
Arvval in Juathuur. Faevv too shows tendencies of this.
Benjamin Prester of A Miracle of Science. Justified, as it turns out: His Science-Related Memetic Disorder is under sufficient control to enable him to hold down a job in the police taskforce dedicated to handling Mad Scientists, but the emotional trauma he suffers when his new partner is attacked and apparently killed pushes him into a relapse.
And in fact, back in his mad scientist days, he not only didn't work alone, but was the only main scientist to build a true alliance with other mad scientists, his friends.
In one of Lazy Muffin's flash animations, "James and the Chief," James says, "I work alone, or with my old partner... Which is dead... So this can't be done."
Spoofed in Homestar Runner, where Strong Bad's Cowboy Cop alter ego Dangeresque claims "I work alone! 'Cept when I work with Renaldo, which is all the time!"
Variant in The Red Panda Adventures: the Red Panda works with a sidekick, but adamantly refuses to work with other superheroes or allow them into his city.
In The Batman, Batman is actually the one to suggest to Superman that he should join the Justice League, but Superman is the one to say "I work alone": An intentional flip of their usual interaction. Cue that annoying gnawing at the back of your head (there's a special shampoo for that). Less than 24 hours ago Batman saved Superman from another kryptonite-laden disaster and Supes is right back to the notion that friends don't help in no time. Guess what the rest of the episode was about.
The Batman, Batman: The Animated Series, and Justice League have splurged this line about a million times each. In one memorable Justice League moment, Batman uses the fact that he's not an official Justice League member to conduct his own investigation when the official founding members decide to turn themselves in to defuse a tense situation.
Bats has the most extended crime-fighting family in The DCU, save possibly the Teen Titans. Nightwing, Robin, Oracle, Batgirl, Spoiler, Huntress, Alfred, the Outsiders, etc. And yet he's constantly trying to sell himself as a lone-wolf-type. At least sometimes he says, "I work best alone."
Darkwing Duck in the pilot and the "Just Us Justice Ducks" two-parter.
Iron Man expresses this attitude at the start of The Avengers: Earth's Mightiest Heroes, when he fights crime out of a personal ambition to keep his weapons out of enemy hands. Soon, four other superheroes help him defeat Graviton. Iron Man subsequently proposes the formation of a full-time superhero team, to combat threats one hero can not overcome alone.
Black Panther enlists five Avengers' assistance in helping him get over the Wakandan border, but insists that he must fight his father's murderer, Man-Ape, by himself. Captain America tags along with Panther anyway, defending Panther from Man-Ape's warriors, while still allowing him to fight Man-Ape alone.
Hawkeye became betrayed by his crimefighting partner, Black Widow, and arrested by SHIELD early on into the series. After he breaks out of jail and clears his name, he decides to chase down Black Widow alone, until The Hulk makes him join the Avengers. Even then, he expresses an initial preference for the other Avengers not to join his personal pursuit.
The title character of Ultimate Spider-Man doesn't initially warm up to the thought of assisting four rookie superheroes as part of his SHIELD training. Unlike most examples though... he has legitimate reasons not to trust them.
In Lego The Adventures Of Clutch Powers, the titular character is The Ace who is used to and good enough to work alone on missions, even when offered help. Naturally, he's given a team after a private adventure in the beginning, and quite a few problems occur because they can't coordinate their abilities.