Area 88: The fighter pilots at the eponymous airbase mix elements of this trope and the Foreign Legion. At one point, they fight the Wolf Pack, who play it straight.
The Lagoon Company itself might also qualify as mercenaries of the modern-day pirate persuasion, although it focuses more on black market goods and personnel courier.
The Extra Order mercs from the first arc are described as war-junkies and take things to Psycho for Hire levels at the Yellow Flag before Revy and the Lagoon Company take them down. They were probably a Shout-Out to the real-world Executive Outcomes, below.
Berserk has mercenaries as its primary characters, with its lead character Guts having been trained as one from childhood. The Golden Age arc of the manga, which the anime covers, follows a mercenary company called the Band of the Falcon that Guts was a part of, and in particular the events that would lead to its idealistic leader, Griffith, undergoing a nasty Face–Heel Turn and becoming Guts' number one enemy.
Bubblegum Crisis: The Knight Sabers build their own Wonderful Toys, but pay for the materials by contracting their services.
Code Geass: After the formation of the United Federation of Nations, the Black Knights become this to them, with Zero as its CEO.
Date A Live: The Deux.Ex.Machina Wizards operate in this way, although they're more like either Murder, Inc. or Psycho for Hire. As the company's private Wizards, they're considerably more powerful and well-trained than the Wizards from AST and SSS. Unfortunately, some of them are just trained to be loyal fanatics and psychopaths who will obey Isaac Ray Peram Westcott's and Ellen Mathers orders without questioning them.
Desert Rose: The protagonists of the manga belong to CAT, which stands for "Counterattack Terrorism". They're a fairly positive portrayal, thanks to the organization's clear mission statement to take contracts to defeat international terrorism and train others to do so. That said, certain chapters show this isn't always clear cut, as the governments they take jobs for may end up being a Banana Republic trying to quash local guerillas — legally correct, if not morally dubious.
Full Metal Panic!: The organization Mithril is one of these, and a large proportion of the show's characters, including one of the two main characters, are members.
The PMC Trust in Gundam 00, a massive coalition of dozens of PMC groups, and powerful enough to have their own dedicated nation. The first season of 00 features two episodes to the Gundams tearing the place apart. Note that the PMC Trust is stated to have an arsenal of almost a hundred Mobile Suits (granted, some of them probably belonged to the AEU who helped defend the place), and Celestial Being only has 4 MS at their disposal. And the Gundams still wrecked the place in just 4 hours...
Serpent Tail in Gundam SEED Astray. In Gundam SEED Destiny Astray, special forces unit of the Eurasian Federation have also deserted to form Mercenary Unit X.
Terminal of Gundam SEED Destiny, which grew out of the original Three Ships Alliance and hired enough new soldiers to become an N.G.O. Superpower on an even greater scale than before. They are an underground armed organization that answers solely to Lacus Clyne, and are not affiliated with either the Earth or ZAFT forces. During the final battle, they get several warships from both sides to defect to them.
The League Militaire (late UC era). They're quite specifically not the Federation military, they just rose up due to the Federation being too apathetic to oppose Zanscare. It's later revealed that one of their primary sponsors is Anaheim Electronics, a major player in the area of Mobile Suit and weapons design, which explains where they get all their advanced equipment.
Mobile Suit Gundam: Iron-Blooded Orphans is centered around one originally called CGS but is renamed Tekkadan early in the show. They both fight and ally with several other mercenary groups over the course of the show.
Hellsing: After most of the members of the eponymous organization are killed, its leader Integra hires the mercenary group called the "Wild Geese", (see under "Film", below) led by Pip Bernadotte, as a replacement Redshirt Army.
Hunter × Hunter: Hunters serve as this, although the Hunters Association is a loose guild — hunters take their own jobs, and the Guild provides contacts and perks.
Jormungand: Several PMCs show up. Koko's bodyguards are on the books as employees of a PMC owned by H&C Logistics Incorporated. The Chinese PMC Daxinghai are the primary antagonists of the first season, and HCLI gets into a fight with British PMC Excalibur in Perfect Order.
SMS in Macross Frontier. They're portrayed in a positive light, and early on are actually more prepared to fight the Vajra than the regular military.
Macross Delta has Chaos, who are unique in that they also operate the Idol Singer group Walkure. This is because Walkure is actually a "Tactical Sound Unit" whose singing is designed to stop a Hate Plague that is itself spread subliminally by song. The series does point out a downside to being in such a mercenary group: mercenaries cannot claim Prisoner of War status and do not have the same rights as actual soldiers when captured.
Naruto: The ninja villages are a fantasy version of this. They're answerable to their nation's daimyo due to their role in national security, but they can contract missions with employers of other nationalities as well. While what their lower-level personnel does can be tasks as mundane as finding cats or doing yard work, the higher-ranked missions often include bodyguards for important political figures, working in place of regular military, or assassination (not that we see them doing the last two parts very much).
The Grass Country Arc was about gathering info from The Mole as a prelude to assassinating Orochimaru, and in the course of which a plot to assassinate Sasuke was uncovered. The following arc revolved around a mission to assassinate Akatsuki duo Hidan and Kakuzu before they captured Naruto. We do see stuff like that, though admittedly these are all security matters and not hired mercenary missions.
Prior to the founding of the villages, shinobi clans more closely filled this role.
This is also what Akatsuki did to raise funds when they weren't pursuing their own agenda. Not that we ever got to see it.
In Negima! Magister Negi Magi, Mana Tatsumiya explained that she was once a part of one of these named Kanbanurae. She traveled to so many battlefields across the far corners of the globe that main lead Negi questioned whether she could actually be 15 years of age. Keep in mind, at the time she was 7-10. Also, Nagi Springfield's group the Ala Rubra used this as a front for their work in the Muggle world.
Obsolete: Cerberus Security Services in Episode 4 provides Exoframe-equipped ex-military personnel to guard an oil refinery in the Persian Gulf.
After the timeskip in One Piece, the Buggy Pirates become this when Buggy, the captain, becomes a Warlord of the Sea. Until after Reverie Arc, of course.
In Sword Art Online: Alicization, the major real-world antagonists of this arc are agents from Glowgen Defense Systems, a powerful PMC based out of San Diego. Real-life San Diego is home to both a U.S. Navy base in Point Loma and U.S. Marine Corps Camp Pendleton; so naturally, many companies that the Department of Defense regularly contracts with have set up in San Diego, such as Kratos Defense & Security Solutions, Cubic Corporation, General Atomics, and local branch offices of aerospace giants Raytheon and Northrup Grumman, among others.
Umineko: When They Cry: In the original sound novels, both the TIPS and Ange mention that along with the rest of Amakusa'slong resume, he also trained Blackwater troops.
Viper's Creed: Arqon Global Security is a PMC tasked with the elimination of "Mech Bugs", stray war machines from a war that ended eight years prior to the story, and still attack cities at random. It is a notable example since the main characters are a pilot and his operator working for Arqon, plus their companions and the overall staff make up most of the cast. Basically the anime is about the PMC.
, such as Kratos Defense & Security Solutions