You can catch Caterpie in the Viridian Forest. A little bit of Level Grinding (just a few levels), and it becomes a Butterfree with the Confusion move. In fact, Yellow Version makes it even easier to obtain, since Butterfree learns Confusion as soon as it evolves (presumably to compensate for the fact that Pikachu cannot harm Brock's Pokémon, so you really almost NEED a Butterfree to beat Brock). Since Confusion is a psychic attack, it's strong on its own, exploits many opponent's weaknesses, and it inflicts confusion status.
You can also can catch either gender of Nidoran west of Viridian City, which is before you enter Viridian Forest, and when it evolves into Nidorino or Nidorina at Lv16, you can use the Moonstone you can find in Mt. Moon after beating the first Gym and immediate evolve it into the powerful Nidoking or Nidoqueen respectively. While the two monsters didn't learn many moves through level ups besides Thrash in the 1st generation, they make up for it in stats and the powerful TM/HM moves both of them can learn such as Dig, Hyper Beam, Double Team, Surf, and several others.
By training the useless Magikarp, which can be acquired the moment you get an Old Rod (or even earlier in the first games, if you're willing to buy the Pokémon from a shady merchant), it will evolve into the incredibly powerful Gyarados at level 20. Even if you don't go out of your way to power-level your Magikarp, it's not hard to make this happen before you or most opponents have anything else that compares.
A similar example from Generation I (when Psychics were still the rocket launcher ofElemental Rock-Paper-Scissors) would be to catch an Abra as soon as you reached Cerulean, level it up until it evolved into the very-respectable Kadabra at level 16, then trade it to a friend and back to make it further evolve into Alakazam. Carnage ensues. Sending it to the Generation II games also allows it to learn the various elemental punches, making it even more powerful.
Back in Generation I, you could get Dig before the second Gym, even though it had the stats of an endgame attack. Better yet, it was super-effective against several Gyms and most Mons could learn it, including two of the starters.
The Game Corner was located early in the game at Goldenrod. If you had the patience and skill then you could earn the coins to buy incredibly powerful TMs and Pokémon. They started out at low levels, but the Daycare (which will raise your Pokémon for you) was just a short walk away.
The second generation (and remakes) also gave out Surf after just the third Gym - with the highest possible power without some sort of drawback (lowered accuracy, recoil damage, or the like). Gold/Silver/Crystal gives it to you about a quarter through the game - giving you enough power to breeze through nearly anything that doesn't specifically resist it. Moves of that power become more standard about 75% of the way through the game, which keeps it from being a true Game Breaker.
There's also "Muscle", the Machop you can get in exchange of a Drowzee in Gold/Silver and HG/SS at Goldenrod City. As early as the third gym, you can get a Fighting-type Pokémon that will grow faster than other Pokés you can catch normally and whose typing and moves will help you against at least three of the remaining gyms in the region. Not only that, you can train Muscle up to level 30 - by which point it'll have evolved into the stronger Machoke - then trade it to another game and back for Machamp and it'll still obey you; and, once you get the Fog Badge from the Ecruteak gym, Muscle can reach level 50 before it starts rebelling. Not even the Elite 4 have level 50 Pokémon during your first playthrough.
Headbutt gets given to you shortly after the second badge, allowing you access to the powerful Heracross and the tough Forrtress via its pre-evolution Pineco.
Want a powerful special attacker early-game? Sprout Tower in Violet City is overrun with Gastly at night. Catch a nice, strong one, and it evolves to Haunter by Level 25, upon which you can immediately trade-evolve it into a powerful Gengar.
The engine-updated Fire Red and Leaf Green give the early-game Caterpie family a massive buff in the Compoundeyes ability, which makes Butterfree's status powders more than 90% accurate, the most accurate status attacks apart from Thunder Wave and Spore.
There's the Pickup ability in Ruby and Sapphire. It gives a 10% chance of acquiring an item after each battle. This item can include a Nugget (which can be sold and turned around into 50 Poké Balls or a bunch of healing items), a Rare Candy (a free level-up), or either a PP Up or a Protein (both stat-boosting items which can be sold for almost as much as a Nugget if you don't want the stat boost). Only slightly less useful is its ability to acquire healing items much more powerful than available until around the halfway point of the game or the best Poké Ball carried in stores (with a much higher catch rate for Pokémon). And it is available on Zigzagoon, one of the game's Com Mons found in almost every grass patch and probably one of the first three Pokémon caught by every player. With a little luck or patience, a player can have an entire medicine cabinet at their disposal to go with a never-ending supply of Poké Balls before running into the first Gym Leader.
In Diamond, Pearl and Platinum, there is Starly, a Com Mon at the start of the game, on the very first route. While it does struggle a little at the first Gym, from that point on, it is almost unstoppable, especially once it reaches its final form and learns Close Combat, a Fighting-type move (on a Flying type, and Fighting/Flying has excellent coverage) with 120 Base Power. Staraptor also has a very high Attack stat, and outspeeds most of the Pokémon in the game as well. Throw in Intimidate after it gets to its second stage, and you have a Disc 1 Nuke which keeps nuking even in the Endgame.
If you have a spare Master Ball from any other game, you can trade it to HeartGold or SoulSilver and catch either Entei or Raikou (or both). If you catch them almost right after they flee from the tower, it gives a real big advantage.
In HeartGold and SoulSilver, by using the Pokéwalker, it's possible to catch Pokémon more powerful than what you would normally have access to.
Even more Pokéwalker abuse is possible if you own two copies of the game. Supposing you have beaten the game on one copy, you have likely unlocked the National Dex, and with it, the later Pokéwalker routes that include decently high level Pokémon from Hoenn and Sinnoh. Now, supposing you start a new game on a new cartridge, you can use the gift option with the Pokéwalker from your original cartridge to gift your new game level 30+ rare Pokémon. And the kicker? They count the new game as their trainer. That's right. You can be running around the second town with level 30+ Pokémon that will always listen to you.
A Pokéwalker can also give you access both to a Pikachu and a Light Ball as early as the beginning of the game. Considering Light Ball doubles Pikachu's Attack and Special Attack, you might be able to train an absolutely unstoppable beast before you win your first Badge.
If you can get your hands on a Jirachi (a Disc One Nuke in and of itself) - which many players did before they even started the game, considering they were handed out for two weeks leading up to HeartGold and SoulSilver's launch - you can unlock a Pokéwalker course called Night Sky's Edge as early as the second town. With a lot of walking and a little luck you can get your hands on a TM for Psychic, one of the strongest Psychic-type moves and the strongest one with no real drawbacks to using it.
In Platinum, as soon as the player has their first Badge, they can do a little backtracking with Rock Smash and, if they're lucky, find a Golbat. At level 10, when under normal circumstances Zubat evolves at level 22. And it evolves to the insanely fast and strong Crobat at max happiness, meaning that the player could have the final form of a three-stage family by the time they reach the second Gym. Oh, and both of Crobat's types are super effective against said Gym.
And in HGSS, you know what's an easy way to max out Happiness in no time at all? That's right, the Pokéwalker! And of course, being a Pokémon game, Zubats are about as hard to find as dirt.
In Platinum, you can get the TM for Earthquake (a Ground-type move with 100 base power, 100% accuracy and no drawbacks whatsoever) after obtaining the second badge. All you have to do is find the hidden entrance to Wayward Cave underneath the Cycling Road and explore the cave.
The hidden part of Wayward Cave is also the only place in the game where you can capture a Gible, a Ground/Dragon-type that will evolve into one of the best non-legendary Pokémon in the game. Gible is also quite strong for an unevolved Pokémon, makes great use of the Earthquake TM you'll find in the same cave you catch Gible in, and knows Dragon Rage upon capture, a move which will always deal 40 HP worth of damage, which will easily defeat most opponents in two hits for quite a while. And the moment Dragon Rage starts to become less effective, it will learn the powerful Dragon Claw move by itself.
By exploiting the Event Mon Arceus in Heart Gold/Soul Silver, you can get one of the Sinnoh cover art legendaries at level 1. While Arceus won't likely obey you, the legendary you get will. Needless to say, this requires likely two games, one of Diamond, Pearl or Platinum, the event Arceus (or the Arceus at Spear Pillar only accessible through hacking), and the ability to trade in Heart Gold/Soul Silver.
In Heart Gold/Soul Silver, the Game Corner returns, the gambling replaced by numbers game Voltorb Flip. If you're good at the game, you can be pretty well off. They have some of the strongest technical machines, though they're more expensive, and a Dratini happens to be available for an amount pretty cheap to those who are good at the game. The Dratini has Dragon Rage, too, an attack that does 40 hp of damage no matter what, which can knock out both of Whitney's Pokémon in just two moves.
In HGSS, it's possible to catch a Heracross in Azalea Town after beating Bugsy without much trouble. Coming off with a huge Attack stat, even by endgame standards, as well as decent bulk and speed, it can be taught Rock Smash through an HM to get decent STAB right off the bat. By the time it learns Brick Break, it'll be able to annihilate Whitney, especially if it has Guts. It also can learn a pretty good variety of moves by early-game standards, such as Aerial Ace and Shadow Claw, letting it take on Morty as well. This also applies to DPPt, where Heracross can be captured after the first gym, which by then you'll also have the Rock Smash HM, though using honey to try to catch one requires a lot of patience.
My Pokémon Ranch has an obtainable Shroomish. Now let's see... Oh wow. Level 45. That sounds nice. But wait! There's more! This Shroomish has Seed Bomb. It also has Spore and Stun Spore to the mix. And it also has... False Swipe? Holy cow! That Shroomish is not just any Shroomish! This is also helpful for catching legendaries at ease!
You can pull the same Pick-up trick with Lillipup in Black and White. The swag is not as high-end, but you'll still be set for the rest of the game if you grind early on. Even better, Pickup can now steal away a one time use item the opponent uses up or thrown at them with Fling, meaning you can snag several good berries in a short amount of time.
Dream World adds a whole new arsenal of nukes to the fifth generation. Right after defeating the first gym and retrieving a Plot Coupon (all of which can be done in less than two hours from the beginning of a new game) you have access to the DW. Through the website you're allowed to catch Pokémon not ordinarily available in the games themselves and who also carry very good abilities. This way you can obtain such things as Bidoof with the incredible ability Moody - making him a Lethal Joke Character, Nidoran male or female with Hustle - which, upon evolving into Nidoking/Nidoqueen, acquire the brutally abusable ability Sheer Force, especially now that TMs can be used multiple times -, Tangela with Regenerator, Taillow with Scrappy, among many others. All this after, again, a mere 1-2 hours or normal gameplay.
In Black 2 and White 2, there are several Gen. V mons you can't obtain until late in the game, but can obtain in the Dream World without any worries. This means you can obtain very strong mons like Sawk, Throh, and Druddigon right off the bat, when you normally wouldn't encounter them in-game until Victory Road and Route 23.
Sawk and Throh in Black and White respectively are catchable as soon as you get to the 2nd gym and have stats that are far beyond anything at that point. Sawk is faster than pretty much everything and demolishes anyone who doesn't resist fighting while Throh with his massive health will be next to unkillable and being fighting types they make Lenora a complete and utter joke and easily crush most of your rival's Pokémon with their sheer power.
The Desert Resort (and, by extension the previous route) in Generation V might as well be called Disc One Nuke Resort. It is a home to some of the most devastating Pokémon the game can offer, such as Darumaka, Sandile, Scraggy, and Sigilyph. A special note can be placed on the evolved forms of Darumaka, Scraggy, and Sandile: Darmanitan is Kyogre's Physical and Fire-type equivalent in terms of immediate power, Krookodile is brutally strong and has amazing abilities/typing to work from, while Scrafty has great defenses on both sides, also has good abilities, and has such a good type match up throughout the whole game that until the Elite Four there are only one or two Trainers in the entire game who have an advantage or resistance against it. To make matters worse, even the Fossil Pokémon, such as the elusive Archen is acquired in the desert. Its almost as if the place is purposefully designed to house as many nukes as possible.
You can also use that Return TM to turn Lillipup, a Com Mon available in literally the same place as the Riolu, into a powerful sweeper. Return is a Normal-type move, so the Normal-type Lillipup (and it's more powerful evolutions) will get STAB damage using it. Also available early in the game is the Silk Scarf item, which powers up Normal type moves when held. This combination can continue to be useful throughout much of the game, killing nearly any Pokemon that doesn't resist Normal type attacks in one or two hits.
Another one is Magnemite in Black 2 and White 2, who can be captured right before the second gym, and at worst, one level away from Sonicboom, which at that point in the game is a 2-Hit-KO move to essentially anything while also having a nice Steel/Electric typing granting it 13 resistances, including a lot of Normal-type moves that you will face early in the game. Unlike Riolu, not only does it start strong (having an acceptable 95 base Special Attack), but it also comes with an acceptable STAB move in Thundershock, and has a well rounded movepool thanks to the Move Tutors and the infinitely-usable TMs. And to take it further, the second gym is extremely weak to Magnemite, and it has overall good gym match ups for the rest of the game. It evolves into Magneton at level 30, which is pretty early, and again into Magnezone at the Chargestone Cave, which is not long after it evolves into Magneton. Forget Riolu; this thing, thanks to its insane availability and utility is widely considered to be the single best Pokémon for an efficient in-game run, up there with the likes of Gen 1 Alakazam.
Also from Black 2 and White 2, there is a catchable Volcarona accessible through a path in the Pokémon World Tournament. The catch? You have the option to go after it as soon as you beat the Driftveil gym. It may be a bit tough to catch, but you can plow through most opponents with ease afterwards. Sure, it will be underleveled and not have quite strong moves (The underleveled one starts with Fire Spin, Leech Life, Gust and String Shot)...until you remember there's a move tutor who can teach it Signal Beam if you have enough Red Shards in Driftveil, and that the TM for Fire Blast is obtainable around 2 towns later. Even better, there's a Gentleman NPC in the Nimbasa City Pokémon Center that gives you 10 Red Shards for free.
Black 2 and White 2 have another in the form of not Pokémon, but cash. In Castelia City, you have access to Castelia Cones, which heal any status ailment and you can buy 12 at a time for just 100 Pokédollars. While this item is pretty fantastic in itself (Full Heals, which do the same thing and are found in Pokémarts, cost six times as much and aren't available until after you win 5 gym badges), there's an NPC another town over who will buy a single Castelia Cone for 2000 Pokédollars, which gives you a 1900 profit per cone. And, since you can buy twelve each day, that nets you a profit of 22,800 per day. This becomes especially useful with the introduction of Join Avenue, which can have a shop (or multiple shops) for training Pokémon. The cost to train your Pokémon up a level is high, but it's still less than the profit you get from the Castelia Cones, and you'd still have enough left over to train a Pokémon's individual stats. And again, this is all before the fourth gym.
Join Avenue itself can also aid in acquiring high sums of money. One of the first Shops you can manage is the Market, which is like a Pokemart but generally sells bundle packs of items for less than its counterpart. If you have access to the GTS, you can join the "Magnemite Stock Market", and trade Magnemites simply for bringing visitors to your Avenue. When the Market levels up enough, it will sell packs of either Lava Cookies or Rage Candy Bars (for about 400-600). The same NPC who will buy Casteliacones from you will also buy these items from you, for 4000-6000 apiece.
Yet another example from Black 2 and White 2 is Pokéstar Studios. It's available after the second gym and is basically this game's version of contests, but it doesn't take a whole lot of skill to get good items. The items include gems such as Moomoo Milk (not purchasable outside of Driftveil), Full Restores (not purchasable until after earning all the gym badges), and Max Revives (not purchasable at all). There's no limit to how many movies you can make, either, so you can rack up a pretty good amount of healing items before you even get to the third town.
After getting a Good Ending movie (successfully complete a movie with a Rental), you can make the movie again with your personal 'mons, and depending on the requirements, get a Strange/Odd Ending. If the requirements are met the film will become a cult classic of sorts, and your fans will give you even BETTER items, such as Nuggets, Big Pearls, Old Gateaux and Star Pieces. Star Pieces can be traded to a guy in Anville Town on weekends, for PP Ups. Anville can be accessed as soon as you reach Nimbasa City, possibly before challenging Elesa's gym.
And ANOTHER area from Black 2 and White 2 full of solid Pokémon for you to use is the Lostlorn Forest, which you can access not long after getting the third badge. Every single Pokémon findable there except Combee makes a great addition to your team: Heracross in Black 2 is just as powerful as in HGSS, though White 2 instead gets the less powerful but still good Pinsir, with equal Attack and Swords Dance to utilize. Roselia has very high Special Attack, can utilize many ways to get the foe poisoned and slam them with Venoshock, and use strong Grass-type attacks like Giga Drain and Petal Dance, then it can evolve into a Pokémon with monstrous Special stats. Didn't get a Sewaddle early in the game? Here you can catch an evolved Swadloon instead. Same with Venipede, who evolves into a devastating Scolipede. Lastly, Petilil and Cottonee evolve into monstrous Pokémon (though Cottonee needs the Prankster ability to really be effective). So if you're short on team members, the Lostlorn Forest will be sure to offer.
There's also Zorua. While acquiring one in Black and White requires a long sidequest involving afateful encounter Celebi, it can be obtained as soon as you get to Castelia City. (In Black 2/White 2, you get it in Driftveil City.) It already starts off with decent speed and good offenses for an unevolved mon, but once it gets to level 30 and evolves into Zoroark, it gets even better. Zoroark has monstrous offenses (including great Special Attack) and great speed, as well as a great movepool. Plus, it gets Night Daze, which has a good chance of lowering accuracy as well as being a pretty strong move in its own right. Its ability is also good for mind games. It's definitely a good idea to train one, seeing how early you can get it.
Drilbur from Generation V is a mild example. At level 31, it evolves from a mediocre Pokemon into Excadrill, a Lightning Bruiser that can be twice as fast when a sandstorm is in effect (or have powered-up Ground/Rock/Steel-type moves) and if you delay its evolution, learns Earthquake at level 33! That's roughly 20 levels before most Ground-types naturally learn it!
At one point during Black 2 and White 2 you're forced to encounter a male Mincinno with his hidden Skill Link ability. Catch him, train him, and evolve him into a Cincinno, and he's surprisingly powerful- Tail Slap, Bullet Seed, and Rock Blast all hit exactly 5 times for lots of damage. STAB Tail Slap alone does 188 base damage note For comparison, that's just slightly weaker than Selfdestruct. And much more practical, too., and the latter two deal 250 if they're super-effective. Pokemon weak to Rock or Grass? Taken down very fast. Substitute or Focus Sash users? They don't stand a chance. Steel-types? Just slap them silly with Wake-Up Slap, which he also learns. And he's damn fast and packs a punch, too. The kicker? He can be encountered before the 4th gym, and you'll have to run into him eventually.
If you use the Memory Link, you can encounter a level 35 Darmanitan that used to belong to N in the Desert Resort. Even in challenge mode, your enemies don't reach that level for another two gyms. The only thing stopping it from being a Game Breaker is it has the rather lousy ability Zen Mode instead of the amazing Sheer Force.
Pokémon obtained in Pokémon Rumble via passwords can be this, as they can one-shot anything that's not a boss.
Want to have some fun? Get the NFC figures that you can use in Pokémon Rumble U, use it before the area that has Keldeo as a boss, and it can kick some butt. But what really takes the cake is the Pre-Order Bonus Toy Pokémon figures Black Kyurem and White Kyurem. Now, it's just a bonus at first glance, but when you scan it, you will eventually realize that Black Kyurem and White Kyurem have strength power in the freaking 2,000s! And that's impossible to get this early in the game! Using Black Kyurem or White Kyurem, any of them can kick any boss square in the butt in one freaking hit! Those Toy Pokémon that don't reach beyond that level will NOT stand a chance against your amazing Black Kyurem or White Kyurem figure. To make things even more awesome, tune up your Pokémon figures using the coins that defeated Toy Pokémon drop, and they'll kick even more butt.
Scatterbug, Spewpa, and Vivillion carry on the tradition of being a fairly powerful early-game Bug-type in Generation VI, even potentially getting Compound Eyes, eventually learns the Fairy-type Draining Kiss which is also a lifesteal attack, and finally gets the very powerful Hurricane which normally has lackluster accuracy but the Compound Eyes ability greatly increases it.
Fletchling carries on like Starly, getting fairly powerful as you teach it new moves, and then it becomes Fletchinder and starts getting STAB coverage in the offensively-strong Fire type. Once you raise it to Level 35 and evolve it into Talonflame, it becomes a swift and hard-hitting fiery bird of prey with a great offensive typing, and it eventually gets access to endgame moves like Brave Bird and Flare Blitz (the latter via Heart Scale at the Move Reminder).
You can catch a Pikachu early on. Not only are both of the first Gym Leader's Pokémon weak to Electric-type attacks, but at Level 18, Pikachu learns Electro Ball, a move that does more damage the higher the user's Speed is than the target's. Pikachu also naturally learns Thunder Wave (a 100%-accurate Paralysis-inducing move) beforehand. Pikachu's naturally high Speed and Special Attack allow it to fully take advantage of this particularly devastating combo - Paralyze the target, which drastically cuts their Speed, and then fire an Electro Ball. Congratulations, you now have a highly-accurate 'Zap Cannon Lite' potentially before the second Gym. Anything weak to Electric-type attacks will keel over, and even neutral-damage targets will get a huge chunk taken out of them, if not a KO. And if you're lucky enough to score one holding aLight Ball...
Not only that, you can find a Thunderstone very early on in the game (on Route 10) and Pikachu learns Thunderbolt at level 29. If you train up Pikachu to the point where it has a good moveset (because, after evolving, Raichu will not be able to learn any new moves naturally), you can have a powerful Raichu before the third gym!
Like Black 2 and White 2, you can score Riolu before the first gym. This time though, there is a mechanic that will assure anything in the undiscovered group (including Riolu) to have a minimum of 3 IVs at 31. If you're lucky with the right stats, Riolu starts proving its uses, especially since you get Return before entering Lumiose City. Then comes Shalour where you get another Lucario with Hasty nature/31 Speed IV, but by this point you may use the Mega Stone on your own Lucario.
Also like Black 2 and White 2, in Gen VI Azurill can be captured fairly early in Route 3. Thanks to the addition of the Fairy type, Azurill's family gains additional resistances including an outright immunity to Dragon-type moves, and Huge Power is a massive boon offensively when it evolves and finally matures into Marill and then Azumarill, giving you access to a strong and bulky Water-type physical attacker that can easily trounce Dragons with Play Rough and smack everything else around with STAB Aqua Tail or Waterfall. Also like Riolu, Azurill is caught as a baby in the No Eggs breeding group, which guarantees 3 perfect-strength IV stats.
Early adoptors of Gen VI got access to a downloadable Event Torchic via Mystery Gift, which comes with the Blazikenite Mega Stone and has the ability Speed Boost. As the badge/level loyalty limit for "outsider" Pokémon starts at Level 30 after you beat Viola and keeps incrementing by 10 with each Gym victory, you'll probably never have Torchic disobey you after Viola unless you Level Grind too intensely.
X and Y also introduced a feature called Wonder Trade, which allowed you to randomly trade Pokémon with anyone in the world. You could get anything from Chespin to Zygarde (not that anyone would be willing to send a legendary) before you even reached the first Gym. Although sending something like Scatterbug or Pidgey will reward you with being hated by someone you will never meet.
Pokémon-Amie is available right from the start, and using it to max out a Pokémon's affection only takes about half an hour's worth of minigames, petting, and tasty treats. Doing so will reward you with a Pokémon that has a chance of shrugging off status ailments, surviving an attack with 1 HP that would've knocked it out, gets critical hits much more often, and gets the same experience boost that traded Pokémon do. With a little time and effort, you can effectively have an entire team of lesser nukes by the time you get to the second town. Isn't The Power of Friendship great?
While travelling to your second gym battle through the Connecting Cave, a little patience will reward you with an Axew. Right of the bat, it will know Dragon Rage, a move that always removes 40 HP from the target, regardless of type resistances. At this stage in the game, where most of your enemies will only have 60-70 hitpoints, Axew will be able to mulch anything that isn't a Fairy type in two hits. Not only that, if you keep it in your party for long enough, you'll be rewarded with a Haxorus, boasting a sky-high attack stat, access to powerful moves like Outrage, and a decent turn of speed to boot. Furthermore, the chance to have the powerful Mold Breaker ability will give you the opportunity to negate a lot of defensive abilities, from Levitate to Sturdy, and even bypasses Wonder Guard!