Rogue Trader, the first edition of Warhammer 40000, wasn't meant to be a competitive tournament game, but more like a skirmish game with RPG elements. So it's not really surprising that it was possible to create hideously unbalanced units. Perhaps the most notorious example is a squad of Imperial Guardsmen armed with grenade launchers firing Vortex Grenades. Sure, the grenades were expensive, but that one squad could basically lay down a set of ten templates that would instantly destroy any enemy they touched. To make it worse, in the first edition a squad was allowed to split its fire between several targets...
Second Edition had its share, some of which were errata'd out in White Dwarf. For example, Wolf Guard Terminators were said to be able to take any combination of weapons and could be built from stock parts with an Assault Cannon and Cyclone Missile Launcher [back in the days when you could essentially alpha strike with every rocket in the launcher; White Dwarf said if such a deadly squad actually existed it would have been included in the fluff text], the Imperial Assassin could in theory be disguised as a Gretchin while wearing Terminator Armour and riding a bike [removed when the Polymorphine Wargear was made specific to Callidus Assassins] and the Strategy Card 'Virus Outbreak' could cripple an entire Ork or Imperial Guard army before a battle even started [the official line was that players should destroy their copy of the card].
A lot of Second Edition special characters were also so powerful the game would end up revolving around them; many of the Third Edition changes were designed specifically to play down the monstrously complex special rules and wargear posessed by such characters.
One of the most overpowered units in Second Edition was the Eldar Guardian (basically Space Elf militia). While their stats were actually rather poor, they had Shuriken Catapults for standard weapons, which before 3rd edition toned down were insanely powerful for their price, so that even Space Marines' power armor offered little protection. On top of that, Guardians were so cheap that an Eldar player could spam huge numbers of them and slaughter the enemy's infantry by the dozen.
Second Edition Tyranids didn't use the strategy card system and instead got a ludicrously overpowered replacement; the Tyranid player could roll for every squad, character and vehicle in the opposing army and inflict annoying or game-screwing effects on a large portion of the enemy army before the battle even started. Results included a squad member randomly becoming a Barbed Strangler blast, and a "Lurker" creature being placed inside an enemy vehicle that would attack the crew if it moved. They also got two special mission cards that instantly nullified their opponent's mission, and one ("Tyranid Attack") also gave them infinite reinforcements and two extra turns.
There was also a chronically unbalanced Nid psychic power called "Psychic Scream" which allowed them to attack every enemy Psyker at once and forced them to roll 2D6; if they rolled over their toughness (likely) they wouldn't be able to do anything that Psychic Phase, and if they rolled over their Leadership they instantly died. It seldom took that long for the Nids to kill every enemy Psyker this way. This was nerfed in later editions.
Chaos got a second codex towards the end of 3rd Edition, and it was one of the most overpowered, with multipleGame Breakers:
Iron Warriors lost the usual 0-1 restriction on Obliterators, allowing them to take nine heavy infantry with Shapeshifter Weapons, and could also take an additional Heavy Support choice such as a looted Basilisk. Back that up with a trio of Defilers and you have an army capable of spewing so many plasma cannons, Earthshaker shells, and battle cannon shots that any opponent is going to have a hard time not getting wiped off the board.
Next up was the notorious Siren Prince/Lord, a Daemon Prince or Chaos Lord with a jump pack/wings/bike (anything to make him faster) and the "minor" psychic power "Siren," which makes the enemy unable to attack him with shooting for one turn. The Prince/Lord would then proceed to zoom forward and summon a horde of Daemons/Chaos Terminators right in the middle of the enemy army without disruption. The only catches were that the power was one of six results you could roll up when you paid the points for a minor psychic power (though the risk could be reduced by paying for several rolls on the table) and that it could be countered by some enemy characters (but not all armies had characters who could do this).
Generally, lots of units were more effective than they had any right to be due to the veteran's skills and daemonic gifts available to the whole army. Tank hunters made Havocs' firepower far more brutal than anything anyone else had. Infiltrate allowed units such as Raptors to charge on the first turn or pop tanks with meltaguns. Sergeants were stronger and had more attacks than any other armies' squad leaders. Daemon Princes were only limited by your imagination in terms of power.
The infamous Dreadaxe + Daemonic Stature + Daemonic Strength combo. Take a Daemon Weapon that excels at killing other daemons, upgrade the model to a Monstrous Creature that ignores normal armor saves, maybe add Daemonic Flight for some speed, and the result is a relatively inexpensive Hero Killer that can deal five or six Strength Six attacks that ignore all saves.
Obliterators and Daemon Princes were in general extremely broken, the first one so much so that they had to release an errated version of the Chaos Codex specifically to nerf the Obliterator so that they could be insta-gibbed by a krak missile, which was followed by another round of nerfs in the next codex to further reduce their stats and weapon options. Daemon Princes likewise lost nearly all of their options, with many more being severely restricted (especially since they are no longer allowed to carry daemon weapons).
The Third Edition Craftworld Eldar supplement was not quite as bad, but it was up there.
The Biel-Tan list allowed you to take any infantry Aspect Warriors as Troops choices. Not scared yet? When thirty Dark Reapers are firing sixty Strength 5 AP 3 missiles with 48" range, with a 2/3 accuracy rate, you will be.
Ulthwe replaced the standard Farseer and Warlock choices with the Seer Council, a squad of 2-5 Farseers and an infinite amount of warlocks. Warlocks also got a new power Augment, allowing them to double the Farseer's range. This, coupled with the fact that any Ulthwe Compulsory Choice Guardian Squad were automatically upgraded to Black Guardians (which had improved Ballistic skill or Improved Weapons skill depending on their profession) for free made Ulthwe armies a force to be reckoned with.
Alaitoc allowed you to not only upgrade your Rangers from good to skull-crackingly awesome Pathfinders, and basically let you Troll the hell out of your opponent. For every squad of Rangers or Pathfinders you took, you got to roll on the Ranger Disruption table, randomly forcing enemy units to come in from Reserve, start the battle pinned, or even giving your unit a free round of shooting at them before the battle started. You could take up to eleven units of Rangers or Pathfinders.
The Imperial Armoured Company list had Leman Russ battle tanks, some of the toughest tanks in the game, or artillery platforms, for every slot in the army. The list was so powerful that it had to include special rules like "Lucky Glancing Hit" to allow enemy infantry a slim chance to stun or shake all these vehicles with weapons that normally would be incapable of penetrating a Leman Russ' armor. The list later reappeared as the Armoured Battlegroup list in Forge World's Imperial Armour.
Quite probably the single most overpowered army of 3rd Edition however was the Blood Angels. For starters, they got every benefit standard Space Marines got with none of the downsides, and often for fewer points. However, the real craziness involved the Death Company, who were totally insane superpowered combat beasts... which were also free, instead coming with other rather insignificant restrictions. The real bullshit however came from the fact that Blood Angels had Rhinos with Over-Charged Engines, making them significantly faster than normal Rhinos. As third edition had no limitations regarding the ability to assault out of vehicles... suffice it to say that the Blood Angels codex was feared and hated for a while.
Due to how skimmers and fast vehicles were handled in the new edition, Eldar Falcon grav-tanks were considered the cheesiest unit in the game. They were almost impossible to destroy as long as they kept moving, thanks to their holo-fields allowing them to re-roll on the vehicle damage table and gaining a 4+ save for moving a small distance. As an illustration, a lascannon (one of the most powerful anti-tank weapons in the game) shot had a 1.2% chance of destroying it. Yes, that's one point two, not twelve. This, along with Harlequin payloads who due to the Rending rule would obliterate most squads they assaulted, meant an Eldar army was incredibly hard to stop, let alone damage.
The Eldar also had the ability to take an entire army on Jetbikes, led by a Seer Council, which was frightfully powerful at both range and assault, as well as almost impossible to hit.
Daemonhunters were designed to counter daemonic units, and had a number of special abilities to do so. One was Sanctuary, which creates a bubble around the caster that no daemon can shoot, move, or see through. While the caster of the ability can't shot or assault, nothing prevents other units inside from doing so. Back when Daemons were one of many units in the Chaos Space Marine army this wasn't a major problem, but after Daemons were given their own codex, dropping Sanctuary on objectives or on some psycannon-toting Purgation Squad made the game unwinnable for the daemon army. Furthermore, as Daemonhunters can be allied with any Imperial force, so any Imperial army could take a cheap Daemonhunter Inquisitor to completely screw over any daemon armies they faced. This lasted until Grey Knights were given a new codex in 5th, but see below for why daemons' troubles didn't end there.
The 4th edition Ork codex was widely considered one of the most overpowered codices Games Workshop has ever produced, and is notable for remaining fairly competitive and winning tournaments from the time of its release at the end of 4th right up until the end of 5th.
Many Ork units were under-costed (at the time) for what they do. The basic Ork boy can be considered a Khorne Berzerker without the Power Armor, but only costs as much as an Imperial Guardsman.
The worst offender was the now infamous Nob Biker army, an army that uses Warbosses to count Nobs as troops, then place them on bikes. While previous editions allocated damage to multi-Wound units with an eye towards removing whole models, 4th Edition randomly distributed hits based on models' equipment. This allowed the Ork player to outfit his Biker Nobz so that each model was different, spreading any lost Wounds around to keep as many Nobz on the table as possible - this on top of a unit that was already very tough due to the bikes, and which always counted as being in cover, and which could take Cybork bodies for another save, and could take a Painboy for another save.
The Burna-Wagon: 1 Battlewagon, 1 Big Mek with Kustom Forcefield and Burna, 15 Burna boyz. With the force field counterbalancing the vehicle's reduced speed to be able to fire, the open-topped transport rules allowed this puppy to stick 16 flame templates on top of each other, multiplying the casualties inflicted to the opposing squad until the other player had to take potentially over a hundred armor saves - more than enough to wipe out anything it targeted, barring the mercy of the Random Number God.
The Big Mek and his Kustom Force-Field gives nearby infantry a weak Invulnerable save and provides cover to nearby vehicles. This wasn't so bad during 4th Edition, when Obscured Vehicles merely had a 50% chance to downgrade any Penetrating Hit to a Glancing Hit, but 5th Edition changed things so that an Obscured Vehicle had a 50% chance to negate a hit. This led to "Killa Kan" lists that exploited the fact that all the Kans in a squadron benefited from a Force-Field if one was in range of it, thereby giving up to 9 mini-Dreadnoughts an even chance to ignore enemy attacks as they rampaged across the tabletop.
Chaos got hit pretty hard with the Nerf bat when they got updated for the new edition, but ironically enough another Slaaneshi psychic power soon caused problems. Lash of Submission allowed players to move an opponent's squad up to 2d6 inches, off of objectives, out of cover, or into tight clusters just ripe for multiple blast weapons,
The Tau's "Fish of Fury" tactic turned two Devilfish transports into a mobile bunker for their Fire Warrior cargo. Enemy fire couldn't target the infantry without going through the hover tanks, but the Tau were at the same time able to fire "under" the very transports that were shielding them. Though firepower-heavy armies could just swat the APCs, melee-oriented forces were unable to hit these floating yet impassible barriers, thus negating the Tau's one great weakness and rendering them assault-proof. The tactic was so abused that many Tau players would denounce it, and one of the changes to skimmer rules in 5th Edition allowed them to be attacked in close combat.
The 5th edition Imperial Guard codex had a psyker battle squad with an ability that dropped the Leadership of one enemy unit by the number of psykers in the squad (to a minimum of 2). This already is nasty enough, but combined with an allied Callidus Assassin from the Inquisition codexes it becomes a real Game Breaker - the Assassin had a template weapon that dealt damage based on Leadership, so by combining the two you get a flamethrower that wounds on 2+ and instakills anything without the Eternal Warrior special rule. Oh, and the Assassin would automatically appear near an enemy squad so it was guaranteed to get at least one shot in. On the plus side, this was probably the best way to deal with the aforementioned Nob Bikers.
Perhaps even more game-breaking were "Meltavets," veteran squads (which were made Troops in 5th edition) able to fire three meltaguns, or seven plasma rifle or pistol shots, from the safety of their Chimera transport for pretty cheap. The result chewed through heavy infantry and light vehicles with ease in an era when large numbers of armour reigned supreme.
Combining 5th Edition Imperial Guard with the old 3rd Edition Daemonhunters resulted in the infamous "Leafblower" army list, so named because it blew opponents off the table through a combination of heinous firepower and neutralizing reinforcements. The fluff-based flying Seer Council Eldar list was one of the few that could compete with it, at least until the Leafblower's ubiquity spawned army lists designed purely to counter it.
The entire Grey Knights codex (created by Matt Ward) has been called this due to insanely powerful units such as...
DreadKnights, standard marines in a walker that acted as faster and cheaper versions of Dreadnoughts before they were reconsidered as vehicles.
Terminators and Strike Squads, the most basic troops available. All have storm bolters, force weapons and psychic powers for +1 strength, spelling death to light infantry and monstrous creatures alike for a mere 5 points more than ordinary Marines (although this problem was present in the previous codex).
Paladins, two-Wound Terminators who used the same wargear shenanigans to abuse the wound allocation rules as the above Nob Bikers, only trading speed for tremendous firepower, better close combat ability and increased resilience. These might have fallen out of fashion like the Nobs had it not been for...
"Psyriflemen," Dreadnoughts with two twin-linked autocannons with psybolt ammunition, the most reliable long-range low-mid armour-buster when the current metagame meant large numbers of light tanks were in almost every army.
As if the above weren't enough to deal with, Chaos Daemons players got to deal with the Grey Knights' "Warp Quake" psyker power, which all their troops could use. Essentially it made Deep Striking near the Grey Knights a death sentence. Did we mention that Daemons had to enter the game by Deep Strike?
With all this, the Grey Knights could kill anything, but had one real weakness - plasma weapons, high-Strength armor-piercing weapons that remain the most reliable way to counter power-armored infantry. Enter the Plasma Siphon, a piece of equipment that rendered any such weapons within 12" of the bearer nigh-useless by dropping their Ballistic Skill to 1. Oh, and an FAQ clarified that the pulse weapons used by the entire Tau army counted as plasma weapons in this case.
You could try and argue that the 5th edition Necron codex (also created by Matt Ward) can be considered a Reconstruction of the Game Breaker trope, since the Necrons are supposed to be literally unstoppable metal horrors. Though very slow (unless in transports), the Necrons are incredibly powerful in the midrange, with Tesla weapons that multiply their attacks, and very resilient, able to come back from the death each phase (not turn, phase). And unlike the Tyranids, nearly every problem in the Necrons' FAQ got resolved in their favour, making many suspicious situations even worse when facing them.
The Blood Angels (can you guess who wrote this one?) are an amazingly overpowered army. Every land vehicle is Fast, with the exception of the Land Raider, but that's okay because it can now Deep Strike. Dreadnoughts count as Troops and Elites, allowing you to take eleven in your army, including the new psychic Librarian Dreadnoughts. Many fans consider them, the Necrons and Grey Knights to be 5th Edition's trifecta of cheese.
6th Edition introduced Flyers into the main rulebook, which became GameBreakers for one of the same reasons Fast Skimmers ruled 4th Edition - melee attacks like Thunder Hammers or Melta Bombs couldn't hit them at all. The other was that the only units that could reliably shoot Flyers were other Flyers, and those with the Skyfire rule... and when 6E came out, the only non-flying model with that rule was the Imperial Guard's Hydra flak tank. This was gradually rectified when the codices were updated for the new edition with dedicated Anti-Air units (or Forge World came out with some), but until then, old armies struggled... and even with updates, bringing insufficient AA left players at the mercy against Flyer-heavy foes.
The Grey Knights could legally take up to six Flyers in a 2,000 point game. Assuming the small number of ground forces didn't get wiped out in the first turn, for months there was no way to counter it.
In the same vein as above is the infamous "Cron Air" list: no fewer than nine flyers and a ground unit that, thanks to a quirk in the rules, is more or less literally invincible due to hiding in the corner behind a purchased bunker that couldn't be targeted so long as it wasn't occupied.
"Scythewing," as Necron flier-spam was dubbed, got around the problem many armies faced when trying to spam fliers due to the fact it could also Spam large number of difficult to draw line of sight to Ground units through Scarabs and a unit of Canoptek Spyders.
The Imperial Guard became insanely powerful in 6th Edition for two reasons: everything in the codex is so cheap that its easy to bury the enemy in firepower and bodies, and it had one of the most durable, versatile, deadly, and cheapest Flyers in the game: the Vendetta. This gunship had the higher level of Flyer armor, three twin-linked lascannons meaning it could smoke any tank/flyer/elite infantry with ease, and was a transport able to drop a squad of meltagun-armed Veterans to finish any fight it started. On top of that the Vendetta could be taken in squadrons, allowing nine to appear in a standard game.
The 6th Edition Chaos Space Marine is generally considered to be relatively well balanced, with the notable exception of the Heldrake. In addition to the general difficulties many armies in early 6th edition have with fliers, the Heldrake is the only flier to have an invulnerable save (well the Dark Eldar flyers can buy one for 10 points), and is has the best front and side armor that a flier is allowed to have, making it the most durable flier in the game outside of Forge World supplements. It also has the "It Will Not Die" special rule, so that even if you managed to damage it, there was no guarantee that it wouldn't be able to regenerate the hull point damage before you could shoot at it again. Further complicating matters was its ability to destroy light transports or slaughter infantry during the movement phase, before firing a weapon that could obliterate any infantry model without an invulnerable save or a 2+ armor save, bypassing cover saves. A FAQ ruling would only make matters worse, giving the Heldrake the ability to fire its weapon in a 360" arc - removing the positioning element that was its only real weakness prior to the FAQ.
The 6th Edition Eldar codex has the Wave Serpent. The Wave Serpent is a dedicated transport that is very spammable, has a special force field that turns penetrating hits from the front and the sides into mere glances in a 2+ and may instead deactivate the shield for a turn to fire D6+1 S7 AP- pinning hits that ignore cover. It also has a Scatter Laser that, after hitting something before, makes every weapon in the model Twin-linked, including the shield.
Even more rage-inducing are units with 2++ rerollable saves such as the Eldar Seerstar or Daemonic Screamer star, both of which are functionally invincible, capable of severely damaging any unit in the game, extremely fast, and can lay down a storm of psychic abilities to improve themselves and harm you.
7th Edition seems to be inviting this by doing away with any semblance of Competitive Balance. Now that anyone can ally with anyone, and "Unbound Armies" can ignore the standard Force Organization Chart, there's nothing stopping players from spamming whatever big nasty units they feel like except their opponents' willingness to play a game with them - a White Dwarf article used a force composed only of Black Legion daemon engines and Tau Riptide and Broadside battlesuits as an example of what the new edition offered.
Added to this is the introduction of Daemonology psyker powers, which allow just about any army (including ostensibly loyalist Imperial forces) to summon units of Daemons, including Greater Daemons. Combined with the new Warp Charge system to generate psychic power, a Chaos Daemons player figured out a way to literally double the size of his army in only a couple of turns (in less than 12 hours after 7th edition was released, no less).
The Battle Missions expansion features a special ork scenario which combined with the Apocalypse rules for force selection, take Ghazghkull and a few warpheadz you could have an ongoing Ghazghkull Waaagh!!! from turn 2 to the end of the game! It is countered by the point that Apoc level lets you take Titans or Baneblades.
Ghazghkull Mag Uruk Thraka personified this trope throughout the editions. Currently it's because of his two-turn buff that makes all Ork infantry very mobile, and grants him the hardest defense to beat in the game (Rolling a 1 on a six sided dice, four times in a row), but only for two turns. The problem is that the average game is six turns.
One of the Tyranids' special formations in Apocalypse had a horribly broken rule called "Out of ammo, Sarge!" which was intended to reduce the effectiveness of shooting at it by discounting all shooting if more than six 6s are rolled. Its actual writing omitted the shooting part and used the word attacks meaning that in melee if more than six 6s were rolled the entire squad lost its ability to punch/slash/stab. This is horrible as most melee units will roll upwards of 20-30 dice. Considering this applies to a blob of 120 'Nids, it considerably unbalanced battles and GW took too long to correct the mistake which was glaringly obvious to any player who had experience with several codices.
Another one for the nids in 5th edition involved the Hierophant, the Tyranid's answer to the Reaver Titan. At the time it was written, the Hierophant's sole psychic power was suppose to give it the incredibly durable armor and a modest invulnerable save (possibly to explain why a giant bug made of chitin and bone didn't collapse on itself when someone tapped it). Unfortunately, when the Tyranid Codex got updated, Warp Field turned from granting a 2+ armor save and a 6+ invulnerable save to granting a flat-out 3+ invulnerable save. Now, Warp Field was designed to protect the extremely squishy Zoanthropes, who really did need the 3+ just so they wouldn't be insta-gibbed by artillery fire the moment the enemy turn started. However, when you slapped it on the single creature with the highest toughness and wound characteristic in the game, as well as the Gargantuan Creature rule, it made the thing completely invulnerable to damage. The Hierophant could literally walk up to a group of warhounds of comparable cost and mop the floor with them without breaking a sweat.
Forge World's creations have been very overpowered in the past. The Elysian Drop Troops list in Imperial Armor volume 4 gave the Elysians the ability to take flying Valkyries as dedicated transports for every single squad. And flyers in this game are almost impossible to kill, as non-AA weapons only hit them on a 6 and all weapons have their range reduced by 12", and nobody in a normal-sized game has an AA weapon with a ghost of a chance of taking out a Valkyrie. Did we mention that they get Vultures (Valkyries that sacrifice transport capacity for more guns) as Heavy Support, and their flyers can mount three to five specialized heavy weapons each? No wonder these guys were so good at taking down Tyranids.
Forge World lists are explicitly only meant to be balanced with lists in the same book, they require opponent permission to use and are never tournament-legal, except in their own tournaments (after all, anyone who pays 960 GBP for a Tau Manta needs to take it out the box sometime).
Shortly before the 6th edition Chaos codex was released, we were treated to a White Dwarf update for Daemons. Aside from introducing the new Slaanesh chariots there were also new rules for some resculpted units. The most notable of these were the Flamers and Screamers:
Both units were now T4, with 2 wounds, a 5+ invulnerable save and the Eternal Warrior special rule, making them immune to instant death. At 23/25pts a model, this is insanely durable, requiring 9 bolter shots to kill a single model in an army where the maximum of 54 of these guys can happily be fielded. This might have still been ok, had this not been combined with how powerful both units where.
GW is trying to tone down players abusing the broken armies by stating that the "Most Important Rule" is to have fun, not to win. Generally Store Managers and a good deal of players look down upon "power gamers", players who play only to win by exploitation. However, to move their products, they inevitably make a few lists composed of the most expensive models, which only exacerbate the situation.