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 (which makes perfect sense, since they were created to fight 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.
The beta of Dawn of War II expectedly had... problems, which required balance changes.
The combat shotguns upgrade of Space Marine Scouts had a chance to somewhere around a 15% to knockback infantry units hit that was likely just annoying in most cases against other infantry squads - the worst that can happen is being knockbacked while a weapon team is setting up or down, quite rarely. Against single-unit commanders, the duration and probability of the knockback occurring combined with the fire-rate of a Scout squad's four members regularly stunlocked commanders to death - because knockback could occur while retreating, it was like the commander couldn't even get away while retreating!
A similar issue happened with an upgrade for Hormagaunts, giving their ranged attacks a similar "chance to knockback" mechanic. Considering their large numbers, a good amount of Hormagaunt squads would stunlock anything that wasn't a vehicle.
The Kommando Nob had a explosive activated on himself that was accidentally had an extra 0, doing 5000 damage instead of 500 damage. Not a big problem in most cases since most individual units don't have more than 500 health anyway and any that do can still usually avoid the pre-detonation wind-up, but an Ork player who was aware to this could use it a few times to kill the enemy's base if they were expecting to lose in the usual victory point department, only requiring the upgrade and fairly cheap cost of reviving your commander twice.
The ability And They Shall No Know Fear granted by the Sergeant upgrade for Tactical Marines make them move slower, be immune to suppression, take less damage, do less melee damage, and frequently Knock Back infantry hit by their melee attacks - though it requires them to do enough damage before a single use of it becomes available. In the beta, this damage requirement was so low it'd easily be available somewhere between every and every other fight and the lowered melee damage drawback did not exist, causing the ability to be something of an I Win button overall for Tactical Marines against pretty much every infantry unit, even when the most expensive melee units in game got close to this generally-ranged-purpose unit. Tactical Marines were also easily capable of having one of the decisively best ranged anti-tank weapons in the game at the same time to let vehicles taste of the hopelessness too. A higher damage requirement made the ability require much more thoughtful use rather than the player thinking "I want to win now".
Dawn of War II had problems on release with Tactical Marines. The mainstay infantry unit of the Space Marines with a variety of upgrades that let them adapt almost any situation... only problem was some of those upgrades would turn a bunch of them from a Jack-of-All-Stats collection of units to a Master of All "blob" with the same kind of unit with one of two weapon upgrades between them as a difference - Plasma Guns gave them effective ranged-fire against anything that walked on two legs, or Missile Launchers could be used which were simply one of the best anti-tank weapons in the game, and when some enemy infantry had the gall to manage to close in on them for whatever reason, the Sergeant upgrade granted them the ability And They Shall Know No Fear which, after doing enough damage (easily obtained when your Plasma Gun Tactical Marines quickly melted just about any infantry), temporarily made them take less damage, be immune to suppression and frequently Knock Back enemies in melee combat to crush those "lucky" infantry's hopes and dreams at being useful. They would do less melee damage too but that was of little solace since the Knock Back rendered trying to stay and fight in melee a definite lost battle, just the Tactical Marines would find it difficult to inflict losses when you reasonably retreat. And if all else failed, the impressive durability of the Tactical Marines for this ranged firepower would generally allow them to take minimal losses at the first sign of trouble. The only noticeable drawback was the upgrades required upgrading the not-too-hard-to-buy second tier - the unit itself was buildable in the first tier to require minimal set-up for abusing them thereafter. After getting to the second tier, proceed to get them the upgrades of not-exorbitant-cost for what they did, attack-move them together, and laugh at your enemies' pathetic attempts to resist annihilation and defeat. The Plasma Gun weapon was Nerfed appropriately by making them do less damage against light infantry units, giving early-game light anti-infantry units or suppression teams the chance to actually deal with Tactical Marines blobs and require more finesse to be countered by the unit (and probably you know, actual combined-arms tactics with other types of units) once the second tier was hit.
The Techmarine's Plasma Gun upgrade has an ability called Overcharge. This greatly increases its fire-rate for a while at the cost of rendering the weapon incapable of firing for a longer while after it was over. Unfortunately the effect of Overcharge against infantry used to pretty much obliterate nearly all of them quickly within its duration save for a particularly enormous mass of them and the Plasma Gun could certainly be bought before so many infantry units were made let alone considering the other units the Techmarine player was sure to have with him because the Plasma Gun upgrade for him was hardly expensive for such an output - not firing for a while was a small price to pay for killing a load of the enemy's army or sending them fleeing for their lives against just one unit. Later Nerfs to the weapon were well-deserved.
Much like their loyalist counterparts, Chaos Space Marines, had some balance issues in early ''Chaos Rising". The idea behind them has was that like Tactical Squads, they could get upgrades that would suit them depending the situation. The problem was that the Mark of Tzeentch upgrade, which gave them Inferno Bolts, was too good for its low cost and gave the whole squad ranged weapons that were effective against all targets. Later patches Nerfed this, rendering to an anti-armor infantry weapon.
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 can learn 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. After he does, only the limited number of supplies prevents him from cleansing whole maps alone. Moreover, he's perfectly capable of killing bosses single-handedly.
Your Marines can easily become grossly overpowered in the Chaos Rising expansion campaign. Cyrus can be upgraded with a "Melee" trait that makes all his explosive accessories energy-using abilities instead of supply-limited abilities. If that wasn't enough, instead of planting just one mine, he can now instantly throw a whole bundle of them which nothing in the game can withstand, be it a squad of Plaguemarines, a tank or a freaking daemon. And if he stays pure, he gains the "Mend" trait, which also lets him use energy instead of supply for healing kits. Similarily Avitus now has unlimited energy-based artillery strikes, and Tarkus - frag grenades which he now throws three at a time, and they can also project auras on the whole squad that enhance its offence and defence respectively. As long as they don't run out of energy, your Marines will be the immortal death machines they are supposed to be, and they won't run out of it, because the Force Commander, in addition to his already great energy-restoring ability now has an area-effect ability that decreases the energy cost of abilities. Here's the thing - that decrease is permanent and it stacks, meaning that after several uses of FC's buff, the abilities become free. As if THAT wasn't enough, you can gain all that sheer OP goodness ''after the first mission'' by rearranging your Marines' skill points.
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.
The Assault Terminators of Dawn of War II, despite being a unit that requires Fury, were heavily OP with the lightning claw upgrade. Vehicles are weak in Dawn of War II, so almost all of what you face will be infantry. Assault Termies get a huge damage reduction bonus against other melee and with the lightning claws, they get multiple high-powered area attacks. They'll utterly shred any enemy melee and with their teleportation or support from fellow marines, they can clear out foes that fight from range. It was so bad that they had to be nerfed by no longer gaining experience, as a squad of Level 1 Assault Termies were almost as vicious as high level Nobs!
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.
Spookums gets different abilities for every type of weapon he equips. Giving him a rocket launcher lets him fire a stream of rockets at the (general vicinity of) the target from an even longer range than his normal attack.
Orks have one ridiculous advantage in the Dark Crusade campaign: unlimited Waaagh banners, which are basically turrets that generate Waaagh and detect invisible enemies. This means that an area that faces regular attacks can end up covered in enough turrets that entire enemy squads are torn to shreds by big shoota fire, buying you plenty of time to build your full army and send it to their base(s) to stomp them.
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.
Later in the campaign, you get to outfit Venom Cannons on Carnifexes, which has insanely long range, area of effect damage, and counters everything. And you can make as many as your population cap allows. Build 4 of them in the final campaign mission and laugh as they proceed to one-shot everything in sight.
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.
Also, the Imperial Guard has an access to Long Range Scanner in the HQ building once the Tactica Control is built. It can scan any area in the map, and if it is used on an area which is covered in Fog of War, it will temporarily clear the fog. This makes it possible for the Basilisk to fire earthshaker rounds without needing to get too close to the foe's HQ.
In Retribution's campaign, a dead unit is refunded entirely. Meaning that it's possible to beat levels just by spamming whichever unit is best suited for the situation (or for proppa orky fun, only sluggas).