In Aqua Knight, mad scientist Alcantara and his servants Zykey and Zycrow are introduced like this. However, it is later inverted when Alcantara turns to be actually a good guy, and his whole purpose all the time was to save Zykey who suffers from a disease that consumes her body gradually.
In Axis Powers Hetalia: The titular Axis, Germany, Italy, and Japan, play these roles much more often than they'd like, and act as a rare example of a (villain) protagonist Terrible Trio. Germany might know what he's doing, but Italy is ineffective at best and a downright liability at worst; Japan is something of a wild card. Regardless, they're nowhere near as ineffective as the Allies.
Though they are part of a larger group, Lust, Envy and Gluttony from Fullmetal Alchemist often function as a Terrible Trio group when it's just the three of them, especially in earlier episodes. The big difference is that they are actually effective in their work.
GEAR Fighter Dendoh has three Gulfer "chips" in human disguise that are always causing minor trouble. They usually get their robots destroyed embarrassingly quickly before Dendoh moves on to a bigger threat.
The Phantom Herb Thieves from the first season of Jewelpet; Mint (the leader, utterly convinced that they are great thiefs), Sage (a tall, thin, cowardly guy) and Aojisho (fat and perky).
The French sisters in Jewelpet Twinkle, Marianne being the assertive leader, Catherine the tall and graceful one and Angelina the short and perky one.
Jewelpet Sunshine has Shouko, Waniyama and Charotte, who are the delinquents of their class. Shouko is the human leader, Waniyama is a short crocodile who's the muscle in the group and Charotte is a bee who doesn't do anything special, but is Shouko's best friend. These three are different from the norm in that they're actually good guys and can be useful in a fight.
Captain Gordon, Defender of Earth! and his sidekicks are stuck in this role for the first eight episodes of the Makai Senki Disgaea anime (because they appear much earlier than in the game).
The Collector Count (a.k.a. Prince Eccentro) and his two female minions (Batch and Gluko) in Mon Colle Knights.
The Grandis Gang in Nadia: The Secret of Blue Water; subverted in that they undergo a Heel–Face Turn relatively early in the series, and continue acting as a collective Sixth Ranger until the end. They also subvert the mold set by Doronjo and her cronies just a little — the tall, lanky guy is strong and the short, squat guy is smart.
Only Tobi, Kisame, and Zetsu are left after Pain is defeated and Konan defects from Akatsuki. Kisame and Zetsu, having known Tobi longer and better than any other Akatsuki member ever did, begin to work as henchmen directly under his command. After Kisame's death, Kabuto takes his place. He's not as obedient as Kisame was though and in all likelihood is more of a Dragon with an Agenda.
A better example of this trope in Naruto would be Jugo, Karin, and Suigetsu, with Sasuke Uchiha as their boss.
The Kuja pirates (pirettes?) Boa Hancock and her sisters Sandersonia and Marigold.
Subsequently Captain Foxy and his two main sidekicks Hamburg and Porche in the anime keep reappearing from time to time to stir up trouble and get rather easily beaten. And for Porche to chase after her "Choppy".
To better drive the point home, Foxy, Hamburg and Porche are somewhat similar to the Doronojo Gang, with Foxy actually using a mecha and Porche's looks being the same as Dronio with blue hair and different clothes.
The most common villain in Overman King Gainer are the idiot trio of Kajenin, Enge, and Japoli. They are far more obsessed with promotions and salaries than actually stopping the Yapan Exodus, and are generally always yelling at each other.
Pokémon: The Team Rocket trio. Jessie often takes the Leadership role, while James and Meowth are the henchmen (though, unlike most of these groups, they introduce themselves as equals). Their boss is Giovanni, who started out giving them orders, then as they kept messing up he essentially gave up on them. He lets them follow Ash around to get them away from him. By now he doesn't even recognize them.
In the first few episodes, Meowth was the leader type while Jessie and James were the henchmen.
As of Best Wishes, the trio has gone into serious mode for an undercover mission. So far, they've stayed away from this trope.
An episode of Pokémon Chronicles (spinoff of the main Pokémon anime) showing one of the trio's early missions shows them to be quite competent as well.
Yes! Pretty Cure 5 has Bunbee's three employees Girinma, Gamao and Arachnea. Even though, Gamao quits early in the series, he appears as often as the other three members. The second division has Bloody, Hadenya and Bunbee, who, however, is the inferior to the two former ones, but he appears as often as the other two. The second division is led by Kawarino.
In Fresh Pretty Cure!, Eas, Westar and Soular are Labyrinth's agent trio. After Eas' Heel–Face Turn, Westar and Soular spend a long time as a duo until Northa replaces Eas and becomes their leader.
The three warriors of the Musk Dynasty, Herb, Lime and Mint. Herb is the leader and initially of the opposite sex of her cronies, though through no choice ofhis. Lime and Mint are the dim-witted minions, one big and strong, the other short and fast. For all their comedic moments, they are all extremely dangerous in a fight.
In the last story arc of the manga you also have Kiima, Koruma and Masara of the Phoenix People (with Prince Saffron as their higher-up). Kiima is the leader of the opposite sex, and a running gag is for her two minions to call her a "old" or "mature" woman, to which Kiima responds by bopping them. They are more competent than your average Terrible Trio though, and, like the Musk Warriors, very dangerous.
Revolutionary Girl Utena has got Nanami Kiryuu's Girl Posse, Keiko, Aiko, and Yuuko. While not really a serious threat, the three are Nanami's devoted lackeys and often do her dirty work, which mostly consists of slapping Anthy and calling her a whore. However, Keiko became somewhat dangerous when she was brainwashed and turned into a Black Rose duelist in her spotlight episode. The three are actually better examples of The Nondescript, as Utena wasn't even aware of their names until Keiko's episode, which was well near the end of the second season.
In Sailor Moon, there are three monsters of the week sent by Zoycite (Zoisite) to kill Neflite (Nephrite). They succeed before they are killed by Sailor Moon and Sailor Mars
And there's a more prominent trio in Super S, the Amazon Trio, Tiger's Eye, Hawk's Eye, and Fish Eye, of the Dark Moon Circus. They do a Heel–Face Turn
Dr Eggman and his two meek droids, Decoe and Bocoe from Sonic X, with "Messenger Robo" Bokkun occasionally getting in on the action.
Tsujido, Makabe and Niihari from Speed Grapher fit the personality and physical requirements for a Terrible Trio. However, they're competent, and a genuine threat to Saiga and Kagura for most of the series.
Sukeban Deka has the Mizuchi Sisters, Remi, Ayumi, and Emi, daughters of a corrupt politician but devious in their own respective rights. On the surface they appear to be a group of glamorous and popular high school students who attend Takanoha High and run their respective cliques, but in reality they are as follows:
Emi, the youngest, is a megalomaniac artist with subpar talent who often has her father bribe judges or steals paintings from other artists to pass off as her own. When she discovers a poor student named Junko is planning on submitting a stunning painting to a art exhibition, Emi steals the painting then has her dad's henchmen kidnap, drug, torture, rape, and then kill Junko over the course of a week. Although it's implied her sister Remi was responsible for that.
Ayumi, the middle daughter, runs a drug and racketeering scheme through the school with the help of her personal bodyguards, the Four Kings.
Remi, the oldest, holds the shimmering veneer of being the White Sheep and the Madonna of Takanoha High, when she is by far the worst of her entire family. She masterminds a scheme that kills 74 of her classmates in a bus crash so they can fill Takanoha with the children of rich parents who might endorse her father's political career, brainwashes her followers into a private army, and has her sisters and father killed off when she deems they've outlived their usefulness in the grander scheme of things.
The Skull Trio from Time Bokan, along with their Expies in the other Time Bokan series, most notably Yatterman. Each Time Bokan show centered around mostly unrelated heroes who travel the world, and occasionally time and space. They are always antagonized by a trio of incompetent villains, who are technically different characters in each series, but look and act so similar that sometimes it's hard to tell the gangs from different series apart. Every anime example on this list, from the Team Rocket trio to the Grandis Gang, was inspired by them. Sometimes they're also implied to be Animated Actors, that is, the same characters playing different parts. It helps that they were voiced by the same seiyuu for more than 20 years!
By far the most popular of these groups was the Doronobo Gang from Yatterman, who grace the page image. They're so well-loved in Japan that when the series was remade in 2008, advertising for the remake focused primarily on them, ignoring the show's actual protagonists. In fact, the sequel series is about their descendants, with the original heroes leaving a legacy of Hero Antagonists.
Kiramekiman inverts this — while the trio is still comedic, as detectives they technically qualify as the good guys, while the heroes of the show are thieves (they have a good reason for their thefts, but they're still breaking the law).
From Urusei Yatsura, the "Spice Girls" — three alien girls, Sugar, Ginger and Pepper — are an especially inept example. Sugar tends to be the leader, but they're all equally dumb. They considers Lum, Benten and Oyuki their rivals, but are ignored by them.
Zoids: Chaotic Century has Stinger and his two henchmen, recurring minor antagonists who try to make up for underwhelming Mecha skills with cunning and trickery. Their plans tend to fall through because the heroes are better at Xanatos Speed Chess.