The Necrons in Dark Crusade. They do not need requisition (the main resource of the game, Necrons will get a a boost to their build times instead when capturing strategic locations), the Necron Warriors are free (although they take a long time to build, and the more Warrior squads you have, the longer it takes to build them), as well as the resurrection ability (which means that destroyed units have a chance to come back to life, which can take the Necrons above the Arbitrary Headcount Limit). The Necron Lord himself may be considered overpowered since when fully upgraded, he can temporarily turn himself into an invulnerable C'tan.
In Dark Crusade, once you destroy an enemy HQ, they lose, regardless of any other units or buildings they have left. The Necron can instantly teleport up to five squads of Flayed Ones any place not covered by Fog of War. Get one unit into the enemy base, teleport in all your Flayed Ones, and wail on the HQ. This works as well in multiplayer as it does in the campaign.
The Necron Lord is a strong melee unit that can teleport and can be equipped with, say, a regenerative ability that reduces ranged damage, the ability to turn himself and surrounding units invisible, can resurrect said units to go above the cap, and can turn into a giant invincible god of death. Which unit do you think is going to be sent into the base as a spotter?
Soulstorm made it even more broken with the Essence Of The Deceiver ability, which turns him into a giant invincible god of deceit (who can summon a fake Monolith and mind control a squad).
The icing on the cake? Once killed, the Necron Lord can be respawned where he died, all upgrades intact. Frequently this means inside the opponent's base, which an upgraded Lord can easily solo.
The mass resurrection alone is a bit of a Game Breaker. If you lose a lot of troops, you can just bring every one of them in a small range back to life. And you can do this even when your populations are maxed out, giving you a larger army than you're normally allowed to have!
The Tau in Dark Crusade, likely because they have very powerful ranged weaponry, making it difficult for the other units to close in. They also have early stealth units, which are effective since the enemy will not have many detector units so early in the game.
In all of the expansions for the original Dawn of War, the Eldar are ridiculously powerful (in the first game they merely had an advantage because they were good at taking down heavy infantry, which both Marines and Chaos relied heavily on). Basically, everything you could do, the Eldar could do better. Their early units were quite good, but their late-game units were just insane; Fire Prisms had better armour than Land Raiders (which are Relic units), Fire Dragons had armour comparable to Terminators, were immune to knockdown and could destroy vehicles and buildings in seconds, all their vehicles except the Wraithlord were fast and could do a jump move, giving them extreme mobility, and with the Avatar increasing population cap they could have the largest force in the game. To make them even more annoying to face, they could build Webway Gates that allowed units to be moved between different gates, and those gates could be made invisible. They also counted as base structures so as long as one remained, the Eldar player's base didn't count as being destroyed. This lead to many games becoming several hour long sessions of "hunt the last Webway Gate".
Eldar are especially effective against Necrons, due to their builder unit having an ability that can lock down an enemy building. All Necron units are produced from their Monolith, so the Eldar can just rush the Necron base early in the game, lock down the Monolith and prevent the Necrons from building anymore units.
In Dawn of War II, the Eldar have invisible bases that they can establish anywhere on the map, a very strong economy, a Relic Unit that boosts unit production, and the ability to put Wraithlords (giant walkers that are highly effective against buildings) on overwatch, with a rally point in the enemy base.
The Eldar are an absolute nightmare to face in high resource games as Fire Prisms, Avatars, grav platform teams, Wraithlords and Wraithguards run amok. The latter are easily the most aggravating of the Eldar units thanks to the absolutely obscene damage they deal to everything while lacking the extremely short range that balances them out on the tabletop. The only thing they conceivably have problems killing are fast cannon fodder melee units like Sluggas, Hormagaunts, and Heretics; and those will be chopped to pieces by shuriken cannons.
In Dawn of War II the Dreadnought in the single player campaign can obliterate pretty much everything with the Assault Cannon Barrage (see the intro of the first Dawn of War for what it looks like): all kinds of infantry, heavy cover, buildings, vehicles, everything. In addition, the Dreadnought has the Emperor's Fist ability, which will annihilate anything reckless enough to engage it in melee. Both of these abilities recharge quickly and don't require supplies. The Dreadnought is also very heavily armored and deals a lot of damage without needing to use either ability.
Sergeant Cyrus may be unimpressive at first, being frail and not too lethal, but once he learns how to plant explosives (capable of wiping out entire squads of heavy infantry and badly damage or destroy vehicles) and snipe (a One-Hit Kill on any non-boss infantry unit, and deals a lot of damage even then) without breaking stealth, only the limited number of supplies prevents him from cleansing whole maps alone. Moreover, he's perfectly capable of killing bosses single-handedly and you might even prefer him to do that, since, though timely and tedious, this way rids you from the need to click-dance the team around the boss and from them being killed all the time.
In Chaos Rising, Cyrus can be upgraded with a "Melee" trait that makes all his explosive accessories energy-using abilities instead of supply-limited abilities, removing one of his only weaknesses.
If he stays pure in Chaos Rising, he gains the "Mend" trait, which also lets him use energy instead of supply for healing kits. This basically makes your squads immortal in most cases.
Thaddeus with Aegis of Fury (an ability to make them invincible for a time after jumping or teleporting), when combined with Chapter's Fury (an ability that, after killing an enemy, there is a good chance that Thaddeus will be able to perform abilities without energy cost or cooldown restraints). This will allow Thaddeus to jump/teleport and kill stuff constantly, while being unkillable. If there are multiple enemies around to kill, Thaddeus and his squad will truly be the Angels of Death.
Terminator Armour in Dawn of War II's and Chaos Rising's campaigns. While you do get a bit more susceptible to things like the dreaded Eldar High Council, you become largely immune or very resistant to almost everything else, all the while retaining most of the benefits you had. The Terminator squads can also use very powerful Terminator-exclusive weaponry.
Imperial Guard Manticores in Retribution are incredibly powerful on some of the smaller maps where you practically can shoot the enemy HQ from your own (nullifying the supposed weakness of the artillery unit because it's practically defenseless up close).
Ork Stormboyz in Dawn of War II can use their Bomba Boyz ability to heavily damage and knock back/down units (even retreating ones!) into the path of your melee units.
In Retribution, Eliphas as a fully fledged Khorne Lord with some points in Nurgle can use Sweeping Doom to one shot armies and emerge with a health bar hardly nudged. He also has the highest raw DPS of all the heroes in the game, especially with a Daemon Sword, which basically means he can solo entire missions.
The Artifact of Ahriman allows Neroth to shoot a volley of Doombolts every other time he makes a basic attack.
In Retribution's Imperial Guard campaign:
Merrick can be upgraded with an autocannon that calls down random artillery strikes, making him capable of annihilating entire units on his own.
Adrastia can be replaced with Stormtroopers equipped with rapid firing melta guns, allowing them to devour vehicles.
Dropping Bern and Adrastia for their honor guard gives more troops, which then Castor can use incredibly well. Hundreds of Guardsmen for a cheap price, being instantly reinforced by the General's troop drops, giving way to endless Beam Spam.
Bern when his Summary Execution is fully upgraded, as it gives a massive damage boost to the entire army while making them invincible, and the recharge rate for it is less than the duration of the invincibility, so as long as you have some spare Guardsmen to execute, you'll have an invincible army that can destroy anything.
Bluddflag, while not as strong as Retribution's other tank heroes, can be this when his WAAGH! ability is fully upgraded, giving a massive damage boost and suppression immunity to all units, fully healing them, reinforcing any friendly squads, bringing in free troops, and restoring all energy, including Bluddflag's, meaning the ability is both free and recharges very quickly. Its effects also increase the more Orks it affects.
In Retribution's Tyranid campaign, the Tyranid Warriors, a durable three-man squad with area of effect suppression that can be upgraded to do anti-everything damage. Furthermore, with a certain piece of wargear, they can have infinite resources.
The Imperial Guard basilisk, especially in Dark Crusade. They are so much better than any other artillery unit in the game. First of all, they have an insanely long range. Secondly, they have the ability to fire earthshaker rounds, which are different from regular shells in two ways: first, they are more powerful, but second, and more importantly, they always hit exactly what you aim at, unlike regular shells, which can miss. Their only drawback is that it costs 200 requisition and 200 energy to fire one earthshaker, but once you've built up your supplies, you can annihilate the enemy's base from halfway across the map. In Soulstorm, they are somewhat superseded by airpower, but are still a very powerful unit.