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All the Rookie Nine trios get this (Shino, Kiba, and Hinata; and Choji, Shikamaru, and Ino). Then you have the older group (Neji, Lee, and Tenten), and Gaara's group (Gaara, Kankuro, and Temari)
Ranma ˝: Ranma (Hero); Ryoga (Best friend/Rival); Akane (love interest of both). Also, Akane loves Ranma but fights with him constantly; she is not in love with Ryoga and only sees him as her best male friend, and never fights with him. Ryoga loves Akane and Ryoga (until he moves on) and Ranma border on Vitriolic Best Buds. And that is the simplest, least psychotic romantic situation in all of the Ranmaverse.
Both Ryoga and Akane are heavily implied to be Ranma's best friends, completing the essence of this trope even more. It kind of makes sense seeing as how they're the ones he talks to the most about his problems or thoughts. At least it explains why fans ship Ranma with Ryoga and Akane so much.
A similar situation is seen with Ranma, Ukyo, and Akane. Ukyo is Ranma's Unlucky Childhood Friend who he thought was a boy way back when. He is shown to still see her as one of his closest friends and often visits her restaurant just to talk or hang out, and they are close enough where Akane feels jealous over the bond of their shared childhood. Akane and Ukyo also get along MUCH better than the rest of Ranma's finances/love interests and can even spend time together without Ranma. As some fans have noted, Ukyo is basically the Ryoga of Akane's life. It's really no wonder that Ukyo and Ryoga also form the most popular OT 3's (Ranma/Akane/Ryoga and Akane/Ranma/Ukyo), especially in the West where Ukyo is much more popular.
To elaborate: Mikado is the founder of the Dollars, Kida is the shogun of the Yellow Scarves, and Anri is the original Saika and the leader of Saika's Children. This makes the trio the leaders of the three rival factions in Ikebukuro. Suffice to say, things deteriorate from there.
Luffy, Zoro and Nami from One Piece. At the start of the series, they are surely this, being the only members that make up the Straw Hat crew, and even in the later chapters they have shades of this - Luffy is the absolute protagonist, Zoro stills seems to be the most important character in the series after Luffy (being his Number Two and all) and Nami is probably also the most important female character on the show.
Trinity: Superman, Batman and Wonder Woman (lampshaded in the first series arc, during which Wonder Woman even mentions her brief flirtations with both Superman — the archetypal ideal husband — and Batman — the archetypal intriguing bad boy.)
Shazam has Billy, Mary and Freddy as the main members of the Marvel Family. Billy and Mary are literally family (siblings, twins in most continuities), so naturally she and Freddy are the ones who tend to be paired together.
The Trio, sometimes known as the Golden Trio, is a popular name for the three main characters of the Harry Potter series. Harry stood up for Ron against Malfoy and Harry told Ron that they need to go warn Hermione about the troll early in their first year. The three were friends ever since.
Feliks, Net & Nika. They met on their first day in new school and stayed together ever since.
The Mystery Kids: Holly (Ego - the most normal one, uses reasoning to solve mysteries), Miranda (Id - prides herself on her ability to crack terrible jokes, most likely to be inspired to a solution), Peter (Superego - collects car registration numbers, loves technology and most likely to use a gadget to find a solution).
Sonny With A Chance: Sonny, Tawni, and Chad fit this dynamic well, but Nico, Grady, and Zora throw it off balance.
Austin & Ally: Kind of: you have the two titular protagonists, but the "Amigos" part is played straight with Dez and Trish.
Wizards of Waverly Place: Another subversion; Alex, Justin, and Harper kinda have this dynamic; but Justin is Alex's brother, and if you count him, you'll have to count Max as well.
Zeke and Luther is another twist to this. The title characters double the Hero role, Kojo and Ginger fit the sidekick roles but are commonly portrayed as antagonists, plus Ginger is Zeke's sister, and her actress Ryan Newman left after Season 2.
iCarly has Carly and her female friend Sam and male friend Freddie. This relationship is very important to iCarly. It is usually the main focus of the show. When two of them fight the other one usually tries to break it up. But then came Gibby...
Buffy the Vampire Slayer has Buffy and her best friends Willow and Xander. When Buffy moved in to town the three of them became quickly became friends, with Willow and Xander helping Buffy fight the forces of darkness. Buffy is the superpowered Slayer around whose duties the show revolves, with Willow being the smart, shy hacker girl and Xander being the outgoing but not too smart Butt Monkey. The early seasons explored a love triangle between the three, with Willow having a crush on Xander and Xander having a crush on Buffy; the three ultimately remained only friends.
Flashbacks on How I Met Your Mother show us that Ted, Lily, and Marshall were this in college. Their dynamic changed when they met Barney a year after college, and Robin a few years after that, eventually transitioning them into a Five-Man Band.
The first three seasons of Smallville have Clark and his best friends Chloe and Pete. They're Clark's oldest friends and the three of them have been friends for years before the series started. They work on the school newspaper together.
Angel: As the name of the show implies, Angel is the main character, who founded the agency which works to combat evil. Early on, he was joined by Cordelia, the ex-spoiled rich girl who still loves clothing and aspires to acting stardom, and Doyle, a half-demon who would just like to fit in somewhere and whose visions of people in trouble let the team know where they are most needed. After Doyle's death halfway through season 1, Wesley joins the team, taking Doyle's place as the male friend. Wesley's thoughtful, caring, bookish nature contrasts sharply with Cordelia's ditzy, rather mercenary personality, but they all manage to get along.
Power Rangers S.P.D. does a fake-out with this trope: The ranger team starts out as Sky, Bridge and Syd, who fit the trope closely, but after Jack and "Z" arrive, they get rounded out to a Five-Man Band by the end of the two-part premiere.
Persona 3: The main character, Junpei and Yukari, at least early in the game before the whole team pulls together into a group of genuine True Companions. Mitsuru, Akihiko, and Shinjiro were ˇThree Amigos! in the early days of SEES, but fractured prior to the events of the game.
Persona 4: Main character, Yosuke, and Chie have this dynamic at first, but it fades away as the team roster grows.
The lowerclassmen (Rise, Kanji and Naoto) may fit too.
Elite Beat Agents: Agent J, Agent Morris and Agent Derek. On other difficulties, J is replaced by Agents Spin or Chieftain, and on the hardest difficulty, they are all replaced by cheerleaders (the "Elite Beat Divas").
Chrono Trigger: Chrono, Lucca and Marle of the 2 girls and a guy variant.
Final Fantasy VIII:: Laguna, Kiros, and Ward; Rinoa, Zone, and Watts; Seifer, Raijin, and Fujin for a villanous version.
Pokémon games after Generation II: protagonist, rival, and love interest/friend.
Legend of Legaia: Vahn, Noa, and Gala are the only three party members in this RPG.
The King of Fighters: Among the slew of teams, one such example is the Japan Team Kyo, Benimaru, and Goro Daimon; or in the case of the "Three Sacred Treasures" Team: Kyo, Iori, and Chizuru.
Rival Schools: Every team in the original game is of this format, with the Pacific team being the closest match to the description. Even when one team consists entirely of adults. Even when one team has a Samus is a Girl situation.
Almost every video gaming webcomic ever seems to revolve around two male gamers and one female, who is usually the girlfriend of whichever male lead is the hero. The typical character dynamic seems to be 'wacky' (read: obnoxious Jerk Ass) guy [The Hero], straight man [second guy] and girlfriend [the woman]. Oddly, the dean of video game webcomics, Penny Arcade, is actually the exception (no girlfriend), so Ctrl+Alt+Del is probably the Trope Maker here.
El Goonish Shive started out this way, with Elliot, Tedd, and Sarah, before the cast expanded.
Slightly Damned has Kieri (an angel), Buwaro (a demon) and Rhea (arguably the main character). Buwaro and Kieri are dancing around a romance.
Woods For The Trees has Jess (Male Border Collie), Reuben (Male Fennec Fox) and Isabel (Female Red Fox).
Avatar: The Last Airbender Season 1: Aang, Sokka and Katara. Slightly unusually in that Sokka and Katara are siblings (and therefore have really similar experiences). Rather typical still in that they have very different personalities and capabilities.
Although the three of them can be seen hanging out, they have a weird dynamic. Kuzco sees Malina only as a love interest, but she is generally repulsed by his personality; meanwhile, Kronk is Yzma's assistant and helps her try to kill Kuzco every week.
Hercules The Animated Series: Hercules is The Hero, always having to save the day. His closest friends are Icarus, the outgoing, enthusiastic, and not entirely sane son of the local genius inventor, and Cassandra, the quiet, contemplative goth girl who gets visions of disasters.