Katekyo Hitman Reborn!: Hibari Kyouya is already a very badass fighter with power level waay above the rest of the characters. He is still getting every single Next Tier Power-Up along with everybody else, staying ahead of the curve.
Kino Makoto from Sailor Moon was already an experienced brawler before she even entered the story, and marked her debut episode by punching out Zoicite. Then she got her Senshi powers...
Minako is an even better example. In the anime it's not really visible, even if her moments as a Plucky Girl hint to that. In the manga, she can One-Hit KO Makoto (done when Makoto had been brainwashed by the Dark Kingdom). In the live action, she provoked a collective Oh Crap in a group of youma by kicking one in a column so hard to break concrete, without transforming!! Not to say she was a slouch in the anime. She leapt onto Usagi's balcony untransformed. She also outran a car after getting her Pure Heart Crystal taken out, even when every other senshi simply passed out from sheer pain and exhaustion.
Akane Tendo from Ranma ½ has gotten this on occasion:
The legendary Super Strength Soba noodles confer herculean strength; after Akane mistakenly ate Happosai's, she was able to lift, toss, juggle, and split in half multi-ton, two-stories-tall iron bells. Unfortunately, they had the side-effect of sprouting whiskers on her face until she took the antidote.
A magical swimsuit made from a jellyfish-like substance, which allows the user to swim like a dolphin —not only negating Akane's Super Drowning Skills, but also giving her a speed and coordination that left Ranma wallowing in her wake.
Darker Than Black stars one. He was known as "The BlackReaper" even before he became a Contractor, and in one flashback he completely owned a Contractor in a straight-up fight using only a choke wire, throwing knives, and sheer Bad Ass ninja skills. He still has the aforementioned skills, but now he can also zap the crap out of people.
Gets taken Up to Eleven when it transpires that these same powers make him a full-blown Reality Warper in the right circumstances.
Mahou Sensei Negima! does this for Ku Fei, who goes from "Martial Artist" to "Martial Artist that can shoot beams out of her hand and has a magic staff" as she gains the ability to use ki around the time the second major arc begins. This is actually noted in-universe by the Meta Guy as being more impressive to watch than the stronger full-on magic users, as she's "closer to being human" and thus you can understand exactly how good she is. It's hard to compare to someone who can blow up mountains, after all.
Scar started out as an Ishbalan warrior priest, who had the strength and skill to take on multiple Amestrian soldiers at once, but after being attacked by Kimblee, he received his brother's right arm, which enabled him to deconstruct anything. And later in the manga, he acquires the power of reconstruction by tattooing his left arm with the designs his brother used to have, enabling him to do alchemical transmutations.
Li Ho from The Law of Ueki was considered one of the strongest competitors, and had taken out a large number of opponents despite refusing to use his ability.
In One Piece, CP9 members Kaku and Kalifa were already amongst the world's top assassins and extremely skilled spies. Then they get Devil Fruit powers, which escalate their badass levels to an... unusually awesome degree. Immediately after consumption, no less.
Zoro and Sanji are two of the greatest warriors of the Straw Hat crew, aside from their captain, but unlike Luffy, they have no Devil Fruit powers or anything else beyond Charles Atlas Superpower. Then they learned Haki.
Ace was already a powerful fighter as a child who could hold his own against adult combatants and regularly beat Luffy (who already had his Devil Fruit) in fights. Then he obtained his fire powers and became one of the strongest pirates in the series.
Sabo was just as competent as Ace when he was a kid. Through only martial arts and Haki training he achevied second position in the Revolutionary Army, surpassing expert fighters like Jesus Burgess and Diamante in a tournament. A tournament that ends with him eating the Flame Flame fruit of his late brother.
Brandon Heat of Gungrave was a great Mafia sweeper who was a skilled hand to hand fighter and a great marksmen. Then, he was killed and become Necrolized. He became much stronger and could take a lot of damage.
Medaka Box: Male Lead Zenkichi was already a Crazy-Prepared martial artist who could handle even abnormals in combat. After the battle with Kumagawa, he gained the Parasite Seeing ability which allows him to see things from other people's perspective. Literally.
Then he becomes even more abnormal, gaining one more ability - Devil Style. Which removes all these coincidences that occur around your run-of-the-mill Main Character.
Miyamoto Haiji from Rosario + Vampire, president of the school's karate club. Like almost all of the school's residents, he's some variety of monster, but he doesn't even bother using any monstrous abilities when he first makes a combat appearance. He just wipes the floor with other monsters using his martial arts; he still hasn't revealed what variety of monster he is.
As it turns out, he's a tengu- a bird demon. And what we've seen for most of the series is his humanoid, weaker form.
Monster Rancher Genki may seem to be Badass Normal at first but is often seen focusing energy in order to attack, as well as transferring it to others and putting it into his attacks. Later on it is shown he can launch Spirit Bombs and, in the last two episodes focus the energy of everyone they know, combining the bodies of several of his friends and reviving the Phoenix.
Suzaku Kururugi may count. Already a Badass Normal (he was a skilled Knightmare Frame Pilot and martial arts prodigy) for most of the first season, he slips into this near the end of the first season and into R2. His most obvious upgrade was having the "Live!" geass cast upon him by Lelouch, allowing him to survive situations that would normally have killed him and allowing him to become much stronger and faster than a normal human being in order to do so. However, it had been hinted throughout the series that Suzaku had some as-of-yet unidentified connection to geass... that was never explained or elaborated upon.
High School D×D has Issei Hyodo, the protagonist is a human who got reincarnated into a devil, who happens to possess a more powerful version of a Sacred Gear called Boosted Gear that doubles his power every 10 seconds. Then, he attains Balance Breaker and is capable of fighting stronger beings and curb stomp weaker enemies. And then he attains the Illegal Triana (Cardinal Crimson Promotion being the strongest form) to the point that Ddraig, the dragon sealed inside the Boosted Gear, states that he's now at Ddraig's full power. And then he becomes even more powerful after having his old body dying at the hands of Samael, the Dragon Eater, and both Ophis and Great Red, two of the strongest beings of the series, create a body for Issei while still having Boosted Gear. If anybody took this trope Up to Eleven, it's him.
In Ayashi no Ceres, Yuuhi Aogiri starts as a competent martial artist (who can cook!) and manages to save Aya more than once with his own abilities alone. However, his sister-in-law Suzumi creates a bandana and two wristbands that lends him part of her power as the Celestial Maiden. That gives Yuuhi partial immunity to other Maidens' attacks (including Ceres) and extra strength while he fights (as shown during the Grand Finale when he manages to severely wound Mikagi with one strike).
The anime of Soul Eater had Maka, the protagonist, a Badass Bookworm who wields a giant red-and-black scythe who is actually her shape-shifting best friend. Her father also had this ability, but she inherited the powers of a Meister from her mother. Until the final episode, when it turns out she has been a Weapon all along.
Satsuki Kiryujin, of Kill la Kill, starts off the series as having no powers besides an extremely dangerous sword and a hell of a lot of training. When the main character, Ryuko, gets her hands on a Kamui and proves nearly able to match her in a fight, Satsuki eventually acquires one of equal power, and starts wearing it for most of their fights. Ryuko herself could qualify, being able to take down superpowered One-Star Uniform wearers even before gaining her Kamui, but she obtains the outfit in the first episode and rarely takes it off thereafter.
Leorio from Hunter × Hunter is the only member of the main cast without nen abilities. While the weakest of the four, he's still stronger than your average human, is a skilled doctor, and displays a large amount of Street Smarts. Then after a Long Bus Trip, he returns with a nen ability that allows him to transmit his punches through the floor, which he immediately uses to lay out legendary hunter and Gon's father Ging.
Vampire Batman. The trilogy of stories ended with everybody dead, including Batman himself.
There are occasions where in Batman has his own Powered Armor to beat superpowerful foes.
Iron Man, who began as a simple man (not even badass) wearing a Power Armor, gave himself technology based super powers with the 'extremis' virus after deciding he wasn't fast enough to keep up. In addition to speeding up his ability to control his suits by mental link (even to summon its pieces and assemble into the full suit with his mind) it also allowed him to mentally link with other pieces of technology as well, often leading to 'I hacked it while standing here talking to you' asspulls.
The new Bleeding Edge suit can do a lot of what the Extremis suit could do and a lot of what it can't.
Ultimate Origins #1 reveals that, apparently, Nick Fury's incredible badass-ness comes from having been a successful test subject of the Super Soldier Serum. On the one hand, you'd think just being Samuel L. Jackson would be enough; on the other, this kinda makes it easier to swallow things like dodging automatic weapons fire in close quarters, or holding his own against Wolverine until Captain America dropped the mutant with a taunt and a grenade. And most of his awesome is in his badass planning, anyway.
This is at least partially true for the normal Nick Fury, who is Bad Ass and... virtually immortal, thanks to an anti-aging formula he took so he wouldn't age out of useful service. That near-immortality applies only to natural causes of death, though; Nick's survival in combat is purely a result of his badassitude.
How to create T'challa, the Black Panther: start with an African prince, raised since his earliest days to be an all-round expert athlete, brilliant mind, skilled warrior and natural leader. Then have him eat a mystical plant and have all of his physical capabilities boosted to explicitly superhuman levels. Season with all of the miraculous gadgets that a super-advanced Mary Suetopia can cook up as personnel-grade gear. Then let the local Panther deity grant him its blessings.
This happened to a huge number of DC Universe'sGolden Age superheroes, either the characters themselves or their legacy successors. Originally, non-powered mystery men were a popular character type back in 40s and all of DC's characters were supposed to be in self-contained continuities. As time passed, these characters were put into a Shared Universe with super powered characters and non-powered characters just didn't seem that impressive in comparison. Since a lot of these characters are fairly obscure, most people didn't really care much if they were reworked. All-Star Squadron was particularly guilty of this.
The most famous example is Black Canary. Before joining the Justice League, Black Canary had no superpowers, but one issue immediately after joining the Justice League, Black Canary suddenly acquired the Canary Cry, a sonic projection weapon emitted from her mouth. This conflicted so much with her traditional Badass Normal nature, that her solo series and the smaller scale Birds of Prey constantly use Kryptonite Is Everywhere to force her to solve problems without using superpowers.
Oh, it gets better. The Canary Cry is sometimes described as being so insanely powerful that any time she uses it, she's basically holding back so as not to completely and utterly eff up everything within earshot. It's put a hurt on characters who are considered to be much heavier hitters. Of course, this ironically lets the writers who are so inclined hold this back and let her use her 'fu as her primary weapon: The Canary Cry really is that powerful, and you use the conventional weapons before you Nuke 'em. Gangs of Mooks and non-Superman-class superpowered thugs just don't rate using it.
The Sandman - Legacy successors had all sorts of super powers.
Congo Bill - Gained the power to turn into a gorilla.
Crimson Avenger - Legacy successor is a spirit of vengeance with magic guns.
Red Tornado - Legacy successors could manipulate wind in the form of tornados.
Blue Beetle - Inverted this and then played it straight. The first Blue Beetle (Dan Garrett) had super powers from a magic scarab, the second (Ted Kord) couldn't make it worknote (Well, sorta. It did offer him its power once, but it turned out to be evil, and Ted chose instead to destroy it. It's back now and doesn't seem evil anymore.), but the third (Jaime Reyes) goes back to using the scarab.
Rex the Wonder Dog drank from the Fountain of Youth, gaining the ability to talk and unspecified magical abilities. Overkill, considering that he was already a successful newspaper photographer before getting the ability to talk.
Averted in PS238 with resident student Badass Normal Tyler Marlocke. When his clone Toby gains incredible cosmic powers his lack of control coupled with no longer relying on ingenuity and determination to survive lowers his badass quotient to almost negligible levels, While Tyler's unpowered alter ego Moonshadow continues to aspire to even greater feats of badassery.
Near the end of Blackest Night, Ganthet triggers an emergency mechanism in the rings of the highest-profile member from each Corps, prompting them to create a temporary duplicate ring and 'deputise' the nearest suitable individual. Most of the resulting Lanterns already had powers (Star Sapphire Wonder Woman, Indigo Tribe Atom, Blue Lantern The Flash), but we also get Orange Lantern Lex Luthor and Sinestro Corps Scarecrow. Unfortunately, it doesn't take long for the Orange Ring to drive Lex mad(der) with greed, and the first ring he steals is Scarecrow's.
Joshua Carver of No Hero did patrols at night without any super powers. Then, he gets FX7 and he becomes very powerful. In fact, he was raised by the FBI as a monster fighter and an FBI agent commented that she's very scared of Joshua since he has superpowers.
In 52 Natasha Irons, after a fallout with her uncle, enlists in the Lex Luthor Everyman Project and gains actual superpowers, going from a human in Power Armor to a human capable of crushingPower Armor. John Henry is infected with a metagene against his will and transforms into a being composed of stainless steel, capable of deflecting bullets and hurling blobs of molten metal.
Tigra of Marvel Comics fame started at The Cat, a lab assistant who used a super suit to make her comparable to Captain America, and give her finger claw grapples (the suit was later taken on by Patsy Walker, Hellcat).
Patsy Walker herself started out as a teen humor character until her transition into a Badass Normal superhero. She later developed some vaguely defined psychic abilities, placing her squarely in this trope.
There is a variety of this in Irredeemable - while Charybdis always had the powers, they were pretty weak. So when he got boosted to Plutonian's level, he has an advantage, because he actually knew how to fight.
Spider-Man villain Tombstone started as an Evil Albino Mafia leg breaker. Soon after his first appearance (in which he got thoroughly trashed by Spidey), exposure to an experimental gas gave him rock-hard skin and super-strength.
Ms. Marvel was first just Carol Danvers, an Air Force military figure, investigating about this new "Captain Marvel" and the Lawson scientist. And there came a day, a day unlike any other, when Marvel Comics wanted to have more female super-heroes, and Carol Danvers was exposed to a Kree machine that turned her into a Flying Brick with Kree powers.
At one point, Chase Stein of the Runaways merged with Vision of the Young Avengers. More recently, during Avengers Arena, he has become the new Darkhawk (which is, of course, ironic, because his predecessor in that role was one of the Runaways' enemies back when they were still based out of the Hostel.)
In a What If? story, the Venom symbiote joins with Frank Castle instead of Eddie Brock. Using his new powers, the Punisher's war on crime becomes more brutal than ever and he succeeds in killing several Marvel supervillains before a team of superheros manages to stun the symbiote, allowing Punisher to tame it by telling it that he'd commit suicide and destroy them both unless the symbiote gives him complete control over it. Punisher then tells the superheroes that he's in control now before escaping.
In the Jackie Chan Adventures fic Queen Of All Oni, Jade gets significantly stronger and more durable (plus a healing factor), along with the normal powers of a Shadowkhan when her Queen persona returns.
Kyon Big Damn Hero does this to Kyon, first making him a Badass Normal via martial arts training in Yuki's error correction spaces. Then in chapter 21 he starts getting cool toys. Initially an extra-dimensional super gun, a watch with body armor, and a very nice coat. Then an AI PDA. Then a beam saber. Then an upgrade to his watch to give him gravity powers. Then another extra-dimensional super gun. Then Asakura Ryoko.
A bit of a meta-example is Jonah Hex. For much of his comics career he got by on his wits and his skills, no matter where he ended up, he always kicked ass and took names. When The Moviewas made, they gave him the power to temporarily resurrect the dead.
Ellen Ripley in the Alien series was a normal human who Took a Level in Badass by the end of Aliens by destroying an entire alien hive by herself, fighting against acid-bleeding parasitic Xenomorphs. Then by Alien: Resurrection her clone Ripley 8, who shares most of her memories, receives some Alien DNA as a result of a flaw in the cloning process. Ripley's own blood becomes slightly acidic, she gets a psychic connection with the Xenos, has reduced empathy and predatory instincts, and is able to shrug off being hit in the face with a loaded barbell.
In Elysium: Max goes from an ex-convict stricken with extreme radiation poisoning to a superhuman with Powered Armor grafted directly into his body. Kruger goes from being a badass Ex-Special Forces sociopath to an augmented badass Ex-Special Forces sociopath.
In Skin Game, we find out that Butters has become Batman of Chicago, combining his grasp of magical theory with Bob's power to create magical artifacts which allow him to, for example, create Mind Fog at will, and skate at almost 30 mph. Still, they're just toys compared to the odds he faces. It isn't until the end of the book, where he becomes the newest (Jedi) Knight of the Cross, that he crosses over into Empowered territory.
Both the Knights of the Cross and Nicodemus's Denarians tend to be dangerous in their own right even without holy swords or evil coins, respectively.
In Talyn by Holly Lisle, the main character of the same name begins with limited use of something called the Hagedwar, a tool which grants the user power restrained only by how well they know how to use said tool. However, as the heat gets turned up, Talyn conveniently learns how to do such extraordinary things with the Hagedwar that she essentially becomes a demigoddess.
Tavi of the Codex Alera is the one and onlyMuggle in a world where absolutely everything runs onElemental Powers. He can't even turn the lights on and off by himself. He is, however, so badass that he easily outmatches more than a few people who have Super Strength or can throw waves of fire around, and often saves the day through sheer Badass Normalawesomeness. But he also becomes less and less normal with time; his bond with Kitai increasingly sharpens his senses and improves his stamina, and at the end of Cursor's Fury, he finally starts to get over his mother stunting his magic and develops some rudimentary furycrafting ability. But since he's short about 15 years of practice, he's unable to utilize it properly until he gets some Training from Hell in First Lord's Fury.
In In Fury Born Alicia Devries is a highly decorated ex-marine, ex-cadre genius of literally superior genetic stock. After spending the first half of the book establishing her as a Badass Normal, she then becomes possessed by a demi-goddess of vengeance and shortly after bonds with the AI of an experimental warship.
Kaladin of The Stormlight Archive starts off solidly Badass Normal, managing to kill a Shardbearer, a feat considered nearly impossible for those without Shards of their own. Then he starts developing Surgebinding abilities.
This is actually a question left intentionally open by the author, and a major source of angst for the character is that he may have had the powers forever but simply been too dim to realize it and use them to protect his fellow soldiers. Many of his epic feats of martial skill and survival sort of read differently on a second read-through when you know in advance that he can sustain himself on stormlight, heal any wound with a handful of spare change, and control his momentum and that of everything trying to hit him. Scenes where an opponent has trouble freeing a weapon from a shield or miss with a projectile weapon get an extra helping of "oh. huh."
A close inspection of the chapter from the perspective of a new recruit in Kaladin's squad mentions something that seems a little odd even on a first read(namely, he seems to be surrounded by/trailing a mist while fighting), and when one knows about his surgebinding it becomes blatantly obvious that he has been using it to help his fighting for a long while. He's clearly still a badass spearman in his own right though. This is proven in the second book, when He kills a chasmfiend when temporarily Brought Down to Normal.
In Snuff it's revealed that due to the events of Thud! Sam Vimes can see in the dark, understand the language of goblins, and can ask the Summoning Dark to tell him anything that happened in darkness.
Lionblaze starts out as a very strong, yet very normal, warrior. Then in Outcast, he starts developing the powers of invincibility, becoming exceptionally strong.
Becoming a Clan Leader works this way. They start out as normal warriors like everyone else note Normal warrior = Major badass, but when they become a leader, they receive nine lives, the powers of StarClan, and any other gift the authors decide to give them.
From The Wheel of Time, both Perrin and Mat. Both were fairly badass even before Perrin found his connection with the wolves, and before Mat had his little episodes with the dagger from Shadar Logoth and the Aelfinn and Eelfinn.
Aviendha as well, as she was an AielMaiden of the Spear before becoming a Wise One apprentice and learning to channel. Even her first appearance involves her and a pair of other Aiel fighting (and taunting) a myrddraal, and she was one of the Aiel who took the Stone of Tear.
Used less subtly in the episode "Upgrades", when the three of them all wear Atanik armbands, granting them enhanced strength, plus Super Speed on a level even The Flash would find respectable. Unfortunately for them but fortunately for the continued drama of the series, the armbands impart their superpowers by injecting a virus...that the human body eventually develops an immunity to. This turns out to be why the Atanik civilization is extinct: they relied on the armbands exclusively for their defense, and once the entire population was immune, they were helpless.
Sam Winchester shifts into this in seasons 4 and 5 when he starts gaining direct control over his psychic abilities. However, he eventually abandons them and returns to being a Badass Normal.
In season 7, we have Ghost Bobby.
In Season 9, there's Dean after he gets the Mark of Cain and the First Blade.
Nick Burkhardt in Grimm, is essentially an average human being with above average abilities in combat. The only supernatural ability he possesses is his ability to see Wesen in their true form.
Fan speculation has gone back and forth on whether being a Grimm provided any abilities besides seeing the Wesen's true forms due to ambiguous portrayals of strength and durability. Starting in season 2, his powers provide a type of immune response that fights certain types of attacks and adapts to compensate or incorporates aspects and resistance to further attacks. So far, he now has enhanced hearing that allows him to fight blind and hear better than the werewolf, and a zombie state that allows him superior stamina without raising his heart rate or getting fatigued and not needing to breath for an unknown amount of time.
Captain Jack Harkness from Doctor Who and Torchwood is this. He starts out a badass former time-agent who can easily keep up with the Ninth Doctor in terms of action. Then he dies and is brought back. Permanently.
Both Mohinder and Ando in Heroes spend the first two seasons struggling to keep up in a world of superhumans. They do okay all things considered — even facing the Big Bad, Sylar, a couple of times and living to tell of it, which many characters with powers can't say. But in Volume 3 they both get powers, letting them hold their own.
This is quite often how Exaltation works in Exalted, but mostly for Solars and Lunars. To explicate: Solars are typically Exalted after performing an act of bravery or a great accomplishment. Lunars are typically Exalted after surviving in the face of overwhelming odds.
World of Darkness: Mirrors has the "Dark Hero" template, which (on top of a few inherent abilities such as a power stat and greater durability and healing) allows one to utilise "Skill Masteries", specialized powers based in mundane Skills. The weakest of these constitute Charles Atlas Superpowers, while the greatest are positively superhuman.
Due to the relatively "fair" balance between supernatural and non-supernatural abilities (mundane skills, for instance, grant essentially the same increase in capabilities as ranks in a Mage's spheres), it is not uncommon for a World of Darkness game to start with a bunch of badass normal "hunters" and then have them gain supernatural templates only after a certain number of sessions.
The Sisters of Battle of Warhammer 40,000 are for all intents and purposes normal humans, albeit trained to elite levels, and equipped with Power Armor, and a Boltgun. Their faith in the God-Emperor though is so strong that it gives them incredible powers on par with psionics (only explicitly not warp derived, therefore without possibility of invoking Perils of the Warp). These used to be so good that many players preferred to design their army specifically around Acts of Faith to make a competitive list, although the sixth edition codex has kicked the entire faction down to low-tier.
There's also the Stormtroopers of the Imperial Guard. For reference, the Imperial Guard are an entire army of human Badass Normals, and the very best of them are inducted into Stormtrooper squads, or the Kasrkin squads, if it's a Cadian army. They are given special equipment not normally afforded to rank-and-file guardsmen, such as carapace armour and plasma weapons, as well as a few minor biological and cybernetic augmentations that increase their strength and speed by a small amount.
This is actually the basic idea behind the Space Marines, really. Start with a prepubescent Badass Normal, then fill him full of bio-mods to turn him into a Space Marine.
While the fluff tends to focus on the ones that were formerly Chaos Marines, Daemon Princes are this trope. Ordinary mortals who become veritable Physical Gods in recognition of their sheer badassery. Like Ax'Senaea the Thrice-Possessed, an ordinary human who consumed three of Slaanesh's Greater Daemons to make herself young and hearty again, and was aiming for number four when Slaanesh made her a Daemon Princess.
From d20 Modern: Take a Fast Hero 5/Gunslinger 10/Sniper 5, already a force to be reckoned with, despite having no magic/cybernetics/mutations.. Give him/her cybernetics implants (D20 Future), Mutations (D20 Future), and/or FX items/Magical inherent enhancements/both (Urban Arcana). Then hope you are not going to face him/her.
In Mistborn Adventure Game, you can spend a bunch of Advancements to manifest Allomantic powers if you don't have them already. Since non-powered characters start with better Attributes and Standings, this can make for a very powerful Hero.
Kyle Katarn from the Dark Forces Saga games spent the entire first game and first 3 levels of the second as just a Badass Normal (during which he bests Boba Fett and takes out at least one Kell Dragon with just his fists). He then learns about his Force Sensitivity and gets a lightsaber.
Final Fantasy V: The party start out as Freelancers who can only attack, defend, and chuck items. By the end of the game, they will probably be Freelancers... who have gained a multitude of unique abilities, magic, knowledge of chemistry, et cetera after mastering the job classes granted to them by the crystal shards.
Final Fantasy VI: Terra and Celes are the only two members with access to magic for the very first portion of the game (in fact, this makes them highly valuable to La Résistance). After meeting with Ramuh, the rest of the party gains access to magicite and start learning magic themselves.
Final Fantasy VII: Vincent Valentine was a Turk, one of Shinra's elite Men In Black who enforced the company's will and guarded their VIPs. And then Hojo got a hold of him and turned him into an unaging shapeshifter.
Final Fantasy VIII: Although the main party themselves don't really count, having been trained to use Guardian Forces and Junctioning, they do turn Laguna and his friends into this whenever they share a consciousness thanks to Ellone. They're already capable, well-trained veterans... but now they have access to para-magic, summons, and Junctions years before Garden made these things familiar to the world.
In fact, for one character, Snow Villiers, he becomes an Empowered Badass Normal twice. It's implied he always knew martial arts, but getting hold of an experimental bulletproof Badass Longcoat which enhanced his strength lets him make the most of his inherent strength and skill. And then, he becomes a L'Cie, granting him magic powers and the ability to summon ice spirits that can transform into a motorcycle, to put the cherry on the cake.
Most of Exile's party in the second Knights of the Old Republic are this - they're already war veterans, trained fighters, spies, bounty hunters, etc. But most of them turn out to be Force Sensitive as well, leading the Exile to train them as Jedi. It's also a common fan theory that Carth Onasi from the first game is an untrained Force Sentitive as well.
Resident Evil 5 reveals that Wesker along with the other Wesker children, though he supposedly was the only survivor, was a subject of Project W, the first and greatest of Umbrella's experiments. He was indoctrinated with Oswell Spencer's beliefs and treated with the Progenitor Virus since early childhood. He was a Badass all along.
Resident Evil 6 gives us two more to add: Sherry Birkin, who possesses a healing factor as a result of her exposure to the G-Virus, and Wesker's Heroic Bastard son, Jake Muller, who possesses enhanced strength, speed, and endurance.
This trope is the origin story of every character in Rift, although the means of gaining powers depend on the faction. On the Guardians faction intro, the narration specifies that you were not chosen because of your faith, but because of your strength.
The protagonists of The Elder Scrolls generally either will become abnormal, or at least can become abnormal (the exceptions are The Elder Scrolls: Arena, where you remain normal, if increasingly badass, for the setting throughout the game, and The Elder Scrolls V: Skyrim, where your abnormality is something you were born with, even if it doesn't have much of an effect unless dragons are about).
The Agent can become a werewolf, wereboar or vampire.
The Nerevarine from The Elder Scrolls III: Morrowind eventually contracts Corprus during the Main Questline. Removing the negative symptoms of Corprus leaves the Nerevarine as The Ageless along with being immune to all other diseases.
This happens to Link in various The Legend of Zelda games. Usually he starts the game as a regular sword-wielding kid, but by the end of the game he's acquired various spells and other magical powers, his energy level has been magically doubled, etc.
You can also do this in XCOM: Enemy Unknown . True, a perfectly ordinary squaddie with a rifle can hold his/her own against the aliens, but later on you can turn them into truly Badass psionic Super Soldiers, Mind Rape enemies to death, conjure up psychic shields or take control of enemies. Enemy Within adds bio-enhancements which allow soldiers to heal miraculously from injuries, jump over buildings or turn invisible, and cybernetic enhancements which allow the soldier to take control of a Mini-Mecha which laughs off firepower and is armed with weapons ranging from a flamethrower to a grenade launcher to a Power Fist.
Depending on player choices, Mass Effect's Commander Shepard can be this by the second game. A non-biotic Shepard is a Badass Normal in the first game (or not, given genetic upgrades are standard issue), but Cerberus adds upgrades in the second. Regardless of class selection, Shepard is given a reinforced skeleton, enhanced senses, tougher skin, increased strength and stamina and virtual immunity to toxins. If you choose a new class with an imported save, it's justified by saying Cerberus' tinkering gave Shepard new abilities. Even if you don't change classes from a non-biotic class, the game lets you unlock loyalty powers, which include a few biotic abilities.
At the start of Baldur's Gate, CHARNAME is just an ordinary teenager with 1 level in an adventuring class. By the end of the first game, he/she is a certified Badass after going toe to toe with Saeverok. By the middle of the sequel? A godling capable of flattening armies and transforming into the Slayer. May or may not ascend to full godhood by the end of the game as the God of Murder.
Some classes in NetHack are definitely Badass Normal to start with. For example, even at low levels the Valkyrie and Samurai are competent warriors who can dispatch most foes with ease, despite not having any magical powers. By the end of the game, the character will almost certainly have several major artifacts, vast magical powers, and the ability to go toe-to-toe with the Horsemen of the Apocalypse, before ascending to become a demi-god. Arguably, this is a bit of a subversion, since the game strongly implies that you are one of the horsemen, War, Brought Down to Normal.
Happens to Amy Rose of Sonic the Hedgehog during the Dreamcasteragames as a result of chasing Sonic. In Sonic CD Amy is pretty much a normal girl who who's running speed is slower than Sonic's walking speed. Then comes Sonic Adventure, and thanks to her Hammer, she becomes very acrobatic, nimble and is able to defend herself effectively once she gets used to fighting. Once Sonic Adventure 2 comes up, she's gotten a bit better, able to run faster and perform basic spindash techniques. However, when Sonic Heroes comes around, she's able to execute nearly every move Sonic can (if not in a weaker form), and is even able to go toe to toe with him when the situation calls for it. This also applies to Big the Cat, who went from being a slow runner with a poor jump who can move cars and posesses impeccable fishing skill to a full-on Power type character, able to run at super speed and spindash just as easily as everyone else.
Dishonored: Corvo was an in-universe Memetic Badass before The Outsider gifted him with a Power Tattoo. His skills with a blade were legendary amongst the city guard, and in game they can occasionally be heard recounting times they saw him in action. Giving him supernatural powers just made him more lethal.
New Guard: What...what do I do if I run into him? Old Guard: Try to make a lot of noise when you die. Then the rest of us might be able to catch him.
Parodied in Shortpacked!!, in which Batman has the ability to breathe in space, based on art in his action packaging which appears to depict him fighting Darkseid in space.
Bun-bun was already one of the most dangerous individuals in Sluggy Freelance. Having him become a godlike personification of holidays took him way over the top.
Also the big revelation near the end of "Oceans Unmoving": Captain Blacksoul is actually Bun-bun riding inside the head of a levitating robot with Super Strength.
Actually, giving Bun-bun superpowerful abilities doesn't seem to change much. The biggest thing that helps him fight battles is sheer attitude regardless.
Years into Dan and Mab's Furry Adventures ' run, we find out that Dan's wings were the outward signs of his 'Cubi heritage (his mother was apparently a powerful succubus, something that apparently everyone knew except Dan). Once his heritage fully kicked in, it granted him assorted new powers that he's mostly still getting the hang of. (His ongoing refusal to stay at the 'Cubi academy and train doesn't help.) Dan was a very successful adventurer already before he semi-retired to Lost Lake and his heritage kicked in.
Everyone who plays Sburb is ultimately destined to become demi-gods in a new universe, but given that many of them were badass before playing, their powers come off as this.
In Axe Cop, new powers are really easy to come by for just about all the characters. Axe Cop himself repeatedly just displays and also gains new powers, the latter including for example being given flight by the comic's writer, and making himself into a magical mummy. But, unlike other characters, he usually forgets about any and all of them soon and acts like a Badass Normal again, until it's time to gain yet another new power.
Higgs from Girl Genius is an interesting example. At first he seems normal, but it is later implied that he is actually one of the Jägergenerals.
In the Global Guardians PBEM Universe, the hero Achilles seems to be a non-powered martial artist in the style of Batman. He's actually a genetically engineered super-soldier cloned from the DNA of Lord Doom, whose strength, speed, and ability to shrug off damage come not from his intensive athletic work-outs, but rather from simple genetics.
Rear Admiral (retired) Sam Everheart, U.S.N., in the Whateley Universe, was already a retired Badass Normal before he took out a team of criminals looting a high-tech research facility and got dosed with nanotechnology. Now Hive is strong and fast enough to deal with the mutants at Whateley Academy.
In Comic Fury Werewolf, analytical players tend to catch Werewolves a lot. Give them a role, and...
This includes any role... including being a Werewolf.
Ranger has had this twice. Once was in Game 9, where he was the Medium Mystic and he developed a Game Breaker strategy. The other was Game 11, where he used his analytical powers as a Werewolf.
One episode of The Batman has Batman borrowing Green Lantern's ring to fight Sinestro. It looked like Batman was going to hand Sinestro a fairly heavy beatdown, but then the ring ran out of power. He hands it back to GL, saying it isn't his style.
People like giving Bats the Lantern ring. In Batman: The Brave and the Bold he nearly puts it on for the sake of an even playing field, to the chagrin of Guy Gardner. Sinestro (before his inevitable Face-Heel Turn) half-heartedly agrees, citing regulation, but comes to a compromise by synthesizing a suit that functions as a Lantern's ring would: on willpower.
A number of Batman: The Brave and the Bold episodes give Batman superpowers. One time, he gained bat-themed superpowers from a magical artifact. Another episode caused a mishap by a villain's laser beam to give Batman Plastic Man's powers.
In yet another, he arrives on an alien planet whose atmosphere gives him Superman-like powers, to the chagrin of the local Badass Normal hero (who happens to also be a Batman). After a while he realizes that he's making the alien Batman feel useless, the way Superman sometimes makes him feel, and in the end he has the alien Batman remove his powers to also negate his weakness to quartz.
Sari Sumdac of Transformers Animated. A Badass Normal in her initial mostly human form, once she gets an upgrade to seeming more machine than human, she gets nifty hand blasters and a processing unit boosted even beyond the Cybertronian standard! Not to mention temporary weapons generation.
Robin himself becomes one when he dons the Red X suit, giving him abilities like teleportation and an array of projectiles. A later episode reveals the only reason he doesn't continually use it is because it's powered by Xenothium, an incredibly dangerous chemical.
In Aladdin: The Series, Aladdin winds up getting possessed by Mozenrath and fighting for control of the body. In the middle of the fight, he blocks Mozenrath's magic with his spirit's own mystical power and wins a Beam-O-War with Mozenrath. Unusually for an Empowered Badass Normal, this is never seen again, and it's heavily implied that he can't use it outside his body.
Macbeth from Gargoyles is an Empowered Badass Normal twice over. During his initial lifetime, he was a highly skilled swordsman and all-around warrior, but had no superhuman abilities. Following a bargain with the Weird Sisters and Demona, he gained virtual immortality, and then at some point between his medieval backstory and first appearance in the present storyline, he also learned sorcery, becoming a Magic Knight.
The Pack are one big mess of these; they start out as a group of TV "heroes" who star in a schmaltzy action series (complete with them fighting the "EEEEEVIL NINJAS!") However, all their physicality is completely real, and when they find out about the Gargoyles they give the Clan a run for their money before getting upgrades. Then, Wolf becomes a Petting Zoo Person, Hyena and Jackal get VERY extensive cybernetic enhancements, and Dingo gets a suit of Power Armor. Only Fox chose to remain not only fully human but also serve out the rest of her jail sentence. Then went straight into the arms of David Xanatos, her future husband, exactly as they planned. It wasn't until much later that it was disclosed that Fox wasn't completely normal herself to begin with.
Owen Burnett gains a stone fist from testing a potion for Xanatos, and he uses it in combat. Turns out he doesn't count: he's secretly been Puck all along.
Zaheer of The Legend of Korra was originally a non-bender, and yet he was considered as dangerous as his teammates, all three of who have unheard of or almost unheard of powers. So dangerous that he was locked atop a mountain in a cell only elite metalbenders could open. When he got airbending, he just became all the deadlier. Commander Bumi, a Retired Badass got the same treatment.