Pikachu is considered the best character in Smash 64. Great range, a Thunder attack that had really high knockback, good throws, the third fastest running speed, amazing recovery (in a game where every other character not named Kirby or Jigglypuff has a terrible to mediocre recovery), and a good spacing projectile (Thunder Jolt). In higher-level play, it only gets worse: skilled players know that the hitboxes of most of Pikachu's aerial attacks, and a few of his ground ones (most notably his forward smash) bear absolutely no resemblance to the actual animation of the attack. Combine this with the phenomenal chaining properties of said aerials and the fact that Pikachu's recovery allows for ridiculous off-stage edgeguarding, and in the hands of a skilled player Pikachu becomes a lightning-fast invulnerable combo machine.
Of course, for those who aren't at such a high level, a single move can go a long way towards slaying the god. Kirby's Utilt◊ comes out faster than almost every other move in the game, including a few jabs, can be chained into itself from 0% for up to 80%, can't be DI'd out of and can frequently link into a forward smash, which at that sort of damage can be lethal.
Melee's top tier characters can be considered game breakers compared to most of the rest of the cast.
Fox is at the very top of the tier list. With his extremely fast movement and powerful attacks that combo in perfect synergy, he is considered the most efficient fighter in the game. His up smash and up aerial are very fast and especially potent killers (both can KO reliably under 100%), but what's even more devastating is his down special, the shine; it's considered the best attack in the game because it's extremely fast (comes out on 1 frame, and can be be cancelled immediately with a jump), has invincibility frames, is very versatile, it has high hitstun and can combo with many of his other attacks, and it's a semi-spike, never mind the fact that it can reflect projectiles (which is supposed to be the point of it, judging by its actual name of Reflector). Keep in mind, though, that Fox is very difficult to play well with.
Falco, who is second on the tier list, is almost as good as Fox. While Fox is the best at killing, Falco is the best at comboing and approaching. His laser is the best projectile in the game because its stun allows him to easily disrupt his opponents. His shine is different from Fox's, as it is optimized to combo with his vertical attacks rather than semi-spike. It's almost as good as Fox's, though. Falco additionally has a very fast spike that his Reflector easily combos into, that can KO as low as 50% when opponents are hit offstage by it. His recovery is the worst in the top tier, but that doesn't stop him from being second best.
Sheik, who is third on the tier list, has very effective juggling combos, a devastating chain throw, a strong, semi-spiking forward-aerial, and a talent for knocking opponents off the stage and keeping them off (using needles).
Marth, who is fourth on the tier list, is there for having fast and hugely disjointed hitboxes on every attack, due to his long sword. Add in the fact that his movement suite is only eclipsed by Fox, a powerful and combo-able spike, an amazing juggle, the best grab in the game, a long, fast and powerful forward smash and the fact that the majority of his moves can hit below the stage, giving him an amazing offensive combo and gimp game and a major tournament threat. His primary weaknesses are his lack of projectile, linear recovery and punishable moves (should they not hit).
Jigglypuff, who is fifth on the tier list, has by far the best recovery in the game, the best offstage offensive game, a talent for surviving spikes and meteor smashes, surprisingly good reach (on forward aerial, back aerial, and side special), and incredible aerial mobility. It also has Rest, a highly risky attack that can KO as low as below 25%, and can be combo'd into.
The Ice Climbers are a bit lower than the other top tiers on this list, but they qualify because they have a technique no other character boasts: Wobbling (Named for tournament player Wobblez who discovered the tactic), the game's one and only infinite combo. It's a bit situational since it requires your partner to be close, but if you grab your opponent and let your partner keep using side tilt, you can rack up way more damage than is ever necessary. Follow it up with a side smash and you practically have a guaranteed KO tactic. Even without wobbling, chaingrabs are a constant threat, as the dual nature of the character allows handoffs that most other characters would need a teammate for.
The biggest game breaker in Brawl is definitely Meta Knight, who is usually either outright banned or handicapped in competitive play to compensate for the fact that he's so much better than every other character in the game. To wit: he has by far the best recovery in the game (his recovery is so far distanced and has so many options that it's nearly impossible to edgeguard him), transcendent priority and disproportionately far reach in nearly every attack, which makes him difficult to approach, a nearly-unstoppable ledge game (which explains the oft-maligned ledge grab rule), excellent killing and edgeguarding/gimping ability, the ability to glide under the stage repeatedly to stall for time, and a series of particularly devastating moves. In particular, up aerial (being the shortest duration aerial in the game and part of the reason his ledge game is so incredible, as well as being an amazing juggler), down aerial (which gets gimps on characters with poor recoveries, and is incredibly fast), Mach Tornado (which is highly damaging and fast, has an incredibly large hitbox that blocks attacks and covers everywhere but above Meta Knight, and is difficult to punish), Shuttle Loop (which is incredibly fast, has invincibility frames, and its incredibly high base knockback allows it to get very early kills when near the blastline and on lighter opponents), neutral air (highly damaging, incredibly fast, and a reliable finisher), down smash (incredible reach, reliably KOs around 125% with the back slice and 140% with the front slice, and is incredibly fast), and down tilt (which is excellent for attacking shields and has a chance to trip, and it can lock knocked down opponents) are particularly painful to deal with. Also, there is a glitch (which was banned even when the character was allowed in tournaments) to become permanently invincible (and invisible) by utilising a control exploit to extend Dimensional Capes. Meta Knight's only non-negligible weakness is his low weight and poor momentum cancelling causing him to die very early. But with how difficult it is to hit and string successive hits on Meta Knight, and his godlike recovery preventing any early kills from edgeguarding, Meta Knight doesn't have any problems surviving. Add to the fact that he can be mastered pretty quickly compared to others and you got a real Game Breaker.
He even gets his own tier, SS Tier. Bear in mind that S Tier is the 'God' Tier normally. Hence the occasional term 'All aboard the SS Meta Knight'.
The Ice Climbers in Brawl are usually considered right under Meta Knight. The main, if only reason, are their chain throws. By exploiting their ability for each Climber to grab the opponent immediately after the other Climber throws them, they can infinitely chain throw any character from zero to death (other than another pair of Ice Climbers). As such, if a player is skilled enough with performing their chain throws, they can literally take a stock from just landing one grab. The Ice Climbers are also horrendously difficult to approach, with desynced Ice Blocks and Blizzards walling opponents, and the threat of losing a stock if grabbed making it very disconcerting for players to approach. The Ice Climbers do have some significant weaknesses though, mainly the fact the Climbers can be separated (permanently for the stock if the easily KO'd Nana is eliminated), where a solo Climber loses their infinite chain throws, their walling capabilities, their recovery, and has their damage output effectively reduced to half. The Ice Climbers also have a few poor matchups, with the few characters that are capable of easily separating them while being able to mostly attack them safely (such as the aforementioned Meta Knight, Snake, and Peach). The Ice Climbers are additionally a strong case of Difficult but Awesome with very few players in the world being capable of taking stocks with one grab (thus the Ice Climbers aren't near as dominant as Meta Knight is in tournaments).
As far as Boss Battles in Brawl go, there are a few characters who stand out among the rest:
Lucas, who is considered the overall best character for this mainly because of his healing move that can absorb certain attacks (such as Master Hand's Bullet Attack and Duon's Pink Laser Attack) which is priceless during a Boss Rush, his powerful attacks such as his PK Thunder and Up-Smash, and the fact that his triple jump hits For Massive Damage and goes through the opponent which also makes a great evasion tactic.
If you're okay with using Stickers, Lucas also becomes one of the most broken characters in the Subspace Emissary. He's already great to use in the adventure mode, but the "PK" stickers turn him into a monster; all but three of his movesnote the forward smash, down tilt and up aerial, all of which can be safely ignored in SSE anyway are boosted by these stickers. Slap enough of them on and watch as this adorable child decimates everything in his path.
Charizard is second on this list because his Rock Smash cuts through the bosses' health bars like a hot knife through butter. Combined with the fact the Pokemon Trainer's Down-B move switch the Pokemon out, it also makes a good evasion tactic against a lot of the bosses' attacks including Tabuu's Off Waves.
Donkey Kong is third here, mainly because of how powerful he is and his great reach. His combination of Spinning Kong and Kong Quake along with the occasional Up-Smash and Forward-Smash tears the bosses up in record time.
Marth is fourth on the list for the fact that he's fast, has a good amount of power and reach, and the most important reason of all, his counter attack! This allows you to dodge and attack the bosses for a nice amount of damage after they try to hit you with a physical attack, which makes most of the bosses a trivial aspect.
There is also the Freeze glitch in Melee that freezes the opponent in place. If Nana grabs the enemy and throws them either forward or up, a little blue spark appears for a single frame (1/60th of a second). If Popo hits forward and B at that exact moment, the enemy will freeze in place and cannot move on his/her own. This allows you to beat them up at your leisure. If you want to unfreeze them, grab them and hit A, then throw them; this will release them from their frozen state. Not only that, if other characters try to use certain throws on Nana, and Popo hits forward stick and B at the right time, he can freeze the enemy and Nana in place as well. While obscenely difficult to pull off in an actual match, the glitch is banned from tournaments for its extreme game breaking properties (such as, guaranteeing victory via time out if the Ice Climbers have equal or greater stocks).
In Brawl competitive play, planking, which is camping at a ledge in order to utilize the ledge grab's invincibility frames, is also a subject of debate. It was resolved by restricting the number of times that the player is allowed to grab a ledge during the match. In most cases, the limit is 50 ledge grabs per match, with Meta Knight's (whose planking is especially powerful) ledge grab limit being 35 or less.
Let's all take a step back and look at the most game breaking item in all of the games: THE HAMMER. Not only does it produce instant KO's 65+% of the time, it causes tremendous damage and causes horrible recovery times to happen. It was made less overpowered in Melee and "Brawl" since some Hammers had their heads fall off, leaving the user wildly swinging a stick that did absolutely nothing. Other players could also pick up the hammer head and toss it at the helpless player; it hits just as hard as if it were still attached to the shaft. There's also a very brief moment where the head stays on however, wherein the hammer works normally. And since everyone will likely be scrambling to pick it up first...
Brawl introduced the much meaner Golden Hammer, which appears on a fancy green, glowing pedestal with its own appearance jingle to notify everyone present that it has spawned. It swings twice as fast as the original hammer and allows the user to hover in mid-air a la Princess Peach until the power-up runs out. Mitigated slightly in that some Golden Hammers are just inflatable replicas that are completely harmless and thus the user is totally helpless, although there's no clear indication of this until the harmless squeak toy makes contact with the other players. You can bet none of them will check for this willingly.
Of all the Final Smashes in Brawl, it's agreed that Sonic has the most broken of them. Supposedly, the attack was designed so that players would have a hard time controlling Super Sonic's incredible speed, but the truth is, Super Sonic can turn on a dime, which pretty much invalidates any evasion strategy by just mashing the control stick left and right through the opponent. As this video notes, a good Sonic player has access to the only undodgeable Final Smash in the game, and even a merely "okay" Sonic player will give you trouble.
Except if the opponent is a master at precision evasion (the side-step, not roll) then unless Super Sonic stays on the spot where the opponent is, the opponent can avoid everything.
Another one that should be noted here would easily be Yoshi's, which was completely unexpected. It combines his special moves from Super Mario World, turning him into a colossal Badass dragon who can fly anywhere on the stage shooting fireballs.