Guts from Berserk is as badass as a human can be, and eventually fights demons. At one point he cuts off his own arm, in the middle of battle, to keep fighting.
How normal Guts still is at this point is up for debate since surviving with the brand gradually made him stronger than any human could possibly be. Not to mention the berserker armor that allows him to fight WAY past his limit. However, Guts vowed to just stay plain old Guts no matter what so he is not likely to use true superhuman powers if he can avoid it.
Reiji Takayama from Witchblade. Stony resolve in the face of an Ax-CrazyPsycho Lesbian with an Artifact of Doom-level power is impressive, but what seals it is that his only words to his assistant in regards to whether they should help The Hero Masane Amaha, or run, were this: "BACK HER UP!!!" He then proceeds to pull out a shotgun and blast away, even managing an almost victory, despite being massively overpowered. Was damn near raped at the same time, but merely stared impassively at the crazy bitch, as if daring her to actually finish the job. Later, he commands an army barely a fraction the size of the force he's fighting and manages to waste a full eighth of an enemy mere mortals should have had a bitch of time even scratching, with weaponry that was unreliable at best. After he led his team to patch a whole hangar of robot tanks knowing they're about to go haywire and started a point-blank shootout with units that gone wild before they finished.
His assistant, Hiroki Segawa, despite being the Sycophantic Servant incarnate, stands his ground with little more than a PISTOL and balls the size of small planets, especially considering said nutcase he was fighting could have killed him effortlessly, also qualifies.
All of the named characters of One Piece without Devil Fruit powers who can fight are this. Mainly the Straw Hats, Zoro, Sanji, Nami and Usopp, as they beat elite fighters with sometimes nothing more than a leg (which is mysteriously burning, at least).
Sanji and Zoro are proud owners of Charles Atlas Superpowers. Usopp's durability is easily superhuman, and he's yet to meet an equal in marksmanship and sniping, but otherwise he's nothing special, getting by on ingenuity and smarts rather than just hitting his opponents till they stop getting up (except for Choo in the Arlong arc). Nami likewise possesses no notable physical abilities, relying on her Clima-Tact weapon to see her through battles with massively powerful superhumans.
As a frame of reference for Usopp's ingenuity, he fights Luffy at one point. Even though Luffy was holding back, that scene makes it clear that Usopp is truly badass; he gave The Hero a run for his money while he was in a freaking full-body cast.
Vivi didn't seem all that impressive next to the Straw Hat crew, but she did make it to a single-digit agent in Baroque Works, armed only with sharp pieces of jewelry on lengths of string...and probably undersold her abilities, too.
Not only did she make it to single digits, but during the climax of the Alabasta Arc, she singlehandedly defeated the Mr. 7 team, while she was only ranked as part of the Mr. 9 team while she was infiltrating Baroque Works.
Mihawk, the world's greatest swordsman, seems to have no Devil Fruit Powers. Likewise, Silvers Rayleigh, the first mate of Gold Roger, is shown swimming at one point (indicating he doesn't have a Devil Fruit) and can fight against an Admiral despite being retired as a pirate for twenty years. Did we mention that the sea he was swimming in was infested with Sea Kings, which are usually hostile and about fifty times the size of the average human?
Shanks is a Handicapped Badass, able to cross blades with Whitebeard and walking away. And before he lost his arm, he was once rivals with Mihawk, the world's greatest swordsman, who had sought him out for fights.
The very first chapter of the manga (the second episode of the anime) has Shanks scare a Sea King (a gigantic sea serpent, which often make humans look like tiny specks by comparison) into fleeing with Conqueror's Haki.
On the Marines' side, there's Luffy's Badass Grandpa, former Vice Admiral Garp. The guy has no Devil Fruit powers whatsoever, but was offered the position of Admiral, a position usually occupied by Nigh Invulnerable people who won the Superpower Lottery, several times, but denied, thinking Vice Admiral was enough.
Zephyr from One Piece Film: Z does not have any powers except for Haki (which everyone Vice Admiral or up possess) and a mechanical arm, which he can fight just fine without. He was an Admiral in his youth, before losing his normal arm, meaning he was in the same position as Garp, but accepted. He is incredibly badass.
Hugue de Watteau from Trinity Blood is characterized as being the only ordinary human in the main cast who can stand his ground and actually persevere in a fight against vampires, a proficiency stemming only from extreme training and fitness and reflexes honed to perfection.
Hugue is not entirely normal (he's got Artificial Limbs). A better example would be Leon Garcia de Asturias aka "Dandylion".
Gareki from Karneval is one; his sharpshooting skills are well-honed and exceedingly better than a top fighter of their nation's ultimate defense organization, Circus. Even when he acknowledges that his abilities are far inferior to that of those who own a Circus I.D., he is able to hold his own with a gun (that he's not actually allowed to have on him) next to the said top fighter in a fight against two psychos.
Jounouchi/Joey from Yu-Gi-Oh! He's the only duelist in the core cast without a mystical Millennium item or reincarnation, but is still a pretty skilled player, regardless of what Kaiba thinks. (The only time he had a MacGuffin on his side would be during the Atlantis/Doma fillerarc.) He's also considered the third best player in the entire world by GX. He would likely have defeated the Big Bad during the Battle City Arc if not for his injuries.
That, and whereas the other prominent duelists are loaded with powerful cards, especially Kaiba, Joey's deck, ostensibly due to him being much newer to the game and a lot poorer for one, is littered with weak cards, along with chance-based cards whose effects can even things out when he gets lucky with the result. (He does get stronger cards throughout the series, most notably Jinzo.) As a result, he really has to fight for his victories, though fortunately for himself, he's one of the most driven duelists in the entire series.
Little Yugi, when up against Atem in the Ceremonial Duel, would be another. Whereas the latter is locked and loaded with all three Egyptian god cards and by this point the ability to will his topdecks with his Millennium powers, the non-magical Yugi takes him and his gods down with much weaker monsters augmented by strategic use of spells and traps, and in spite of Atem's ability to select what he draws every turn, Yugi manages to stay one step ahead, culminating in his calling out Atem's final strategy of revive Osiris. (Added in the anime version was Kaiba's presence, where he offers Yugi to choose among his rarest and strongest cards for help against Atem, but Yugi chooses to rely on his own cards. Not only that, he takes down all three god cards at once!)
To a lesser extent, you also have Mai Valentine; not as major a character as Joey, but she was skilled enough to make it to the finals of both Duelist Kingdom and the Battle City tournament. We only see her duel twice in the latter, but she defeats Jean-Claude Magnum pretty easily, and engages Marik in a duel, and fares exceedingly well against him. She only loses due to a trick of the Winged Dragon of Ra unrelated to dueling. She'd have beaten him handily without it. Atem and Kaiba were both exceedingly impressed with her.
Crow from Yu-Gi-Oh! 5Ds was arguably badass right from his introduction. However, his defeat of a Dark Signer (even when faced with a Jibakushin), despite not being a Signer himself, proves that he is definitely qualified for this trope. He also has the series's only Riding Duel First-Turn Win under his belt, and has survived a day and a half inside an old fridge.
His status as such was pretty short-lived, however: with the Second Season Finale, the Crimson Dragon, probably wondering why it didn't choose him in the first place, has given him the Tail Birthmark and made him into a Signer, which sadly means he can't be classified as a Badass Normal anymore.
Nenene Sumiragawa from the Read or Die TV series is one of the only normal humans in the series. However, she is the strongest-willed, and dealing with her leads to the fall of not one, but two Ancient Conspiracies. As a popular and talented author in the world of the series, her ability to write is itself something of a "superpower".
In both the OVA and TV series, Drake Anderson is able to defeat several superpowered beings using only his army training and machinery.
Mireille Bouquet is one of the best assassins in the world, but compared to the rest of the cast of Noir is mundane. However, her refusal to bend ultimately brings down an Ancient Conspiracy (or at least, one branch of it).
Natsuko Aki in Re: Cutey Honey inspires Cutey Honey to fight, after laying siege to a villain's hideout with just the artillery on her back and sheer guts.
The "Nanaya Shiki" of Melty Blood. He's kinda like a fake copy of the protagonist and therefore does not possess the eyes that can potentially kill anything. Doesn't stop him from doing what he does best though.
Kiyone in the Tenchi Muyo!! Tenchiverse timeline had none of the exotic magical or Sufficiently Advanced powers enjoyed by most of the main cast, or even her partner Mihoshi's consistently extraordinary luck, but still managed to at least register on their scale thanks to great skills with her array of high-tech weapons and gadgets.
Nicholas D Wolfwood of Trigun. At least in the anime, he was just a regular if not well trained guy with a really big gun. Despite this he's able to go head to head with large groups of angry thugs, a killer robot and was at least Vash's equal in a quick draw tournament. Averted in the manga where he gained his abilities through a chemical regimen.
Integra Hellsing is fast and smart enough to hold off superhuman foes like Anderson and Nazi vampires attacking from the front despite being a fully un-powered human. Plus shooting a perfect cross of blessed silver bullets into the forehead of a moving vampire. With a Walther PPK. At 20 yards. In a crowded conference room.
Captain Pickman, a minor character only featured in the TV anime and not until the very last episodes, was the last of the Hellsing Redshirts when they were led into an trap at the Tower of London. Despite every single other Redshirt having been violently killed, he continued to fight at the side of Alucard with surprising success until a enemy offscreen mortally wounds him, and requests Alucard to kill him so he would die as a human.
Genjyo Sanzo in Saiyuki is the only full human in his quest party and his main weapon, a small handgun, seems to be devoid of special magic. DO NOT MISTAKE THIS FOR BEING A PUSHOVER. He is an extremely good shot, a skillful strategist, and could best adult men at hand-to-hand martial arts by the time he was 14. They don't give you a sutra just for chanting pretty words.
Togusa from Ghost in the Shell stands out less for being a badass than for being normal. He got selected for one of the worlds most elite counter-terrorism made up from cyborgs as a simple street cop with almost no augmentations except for the ubiquous Brain/Computer Interface. While not being able to keep up with the other members of the team in foor chases or hand to hand combat, he was requested specifically by the units leader, though she never told anyone why. Though he does do a very good job at reminding the cyborgswhy they are fighting terrorism and corruption in the first place.
It is worth noting that while it's never specifically stated as the reason, it becomes apparent that Togusa provides a different perspective for the group, especially in his detective training (the Major mentions offhandedly as much in the first episode). In the first season, he's the one who makes most of major breakthroughs logically in the case.
Major General Olivier Milla Armstrong in Fullmetal Alchemist is the older sister of Alex Louis Armstrong. She's also a sword-wielding maniac who terrifies the Elric brothers. Even though she has no alchemic abilities, and relies only on her sword, she has put more fear in them than the Homunculi, Scar, and Father combined.
And her whole garrison. Who did most of the work of taking over the country toward the end of the series.
Lt. Colonel Hughes is also badass normal to a particular degree, wielding push knives with deadly skill and holding his own against two Homunculi until a third one kills him.
Mustang's men are this trope personified, although special mention belongs to Riza Hawkeye. When the single most deadly alchemist in the entire military hand-selects a Gunslinger Girl to be his bodyguard, you know she's got to have some serious skills.
Sig proves that his muscles isn't just for show in the last arc when he kills Sloth alongside Alex Louis by throwing him on a spike.
Ling Yao before becoming the new Greed is quite the badass. While he has no knowledge of alchemy, he's quite skilled with his sword and at one point holds his own against Wrath while holding an injured comrade across one shoulder. His bodyguards, Lan Fan and Fuu, aren't too shabby, either.
Wrath is the only one of the Homunculi who doesn't possess their Healing Factor, nigh-immortality, or any real superpowers, yet he still manages to be one of the most dangerous characters in the entire series.
In a world containing spirits, gods, and a warrior-prince with a possessed super-powered arm, San is just a girl with a spear, knife, and wolf...who can scale 30-foot walls.
Lady Eboshi, as a female leader in a patriarchal society, only has intelligence and charisma on her side. By the end of the movie, she's taken on everyone up to and including the god of the forest, and he's the only one who got the best of her. And she survives that confrontation.
Ein and the adult version of Bat from Fist of the North Star. While they don't have same martial arts training as Kenshiro and other martial artists in the series, they're capable on holding their own against the bad guys. The two of them working together managed to punch a fortress filled with goons to death. Some of the bad guys, like Jackal, may qualify for this trope as well.
Kai, David, and Lewis from Blood+ could fit the bill, seeing as they are ordinary humans who regularly wade into battle with powerful Chiropterans armed only with simple firearms (which the series establishes as being next-to-useless against them), and emerge almost unscathed every time. Out of all them, Lewis and Kai probably emphasize the "normal" aspect the most, since one is a fat, awkward slob, and the other is an ordinary teenage boy. David definitely fits the "badass" part, since he survives being shot, getting slashed by monsters, and jumping out of a helicopter moments before it exploded, despite requiring realistic medical treatment in each case, and when he infiltrated a secret research facility and found himself surrounded by armed guards, he escaped by grabbing a hostage and threatening to kill him with a ballpoint pen.
Kaneda himself is another example. Anyone who can face down a superpowered psychic psychopath with nothing but a laser rifle without getting splattered definitely qualifies.
In a school inhabited by super-powered Magical Girls, Haruka Suzushiro from Mai-HiME counts, if only for the fact that as the head of Fuuka Academy's Executive Committee, she has a whole legion of students at her direct command, and her utter refusal to take "no" for an answer even when faced with death after Yukino's CHILD is instantly destroyed by a freshly Face Heel Turned Shizuru, whom she chewed out and gave a very nasty headbutt just before she disappeared.
The most exemplary moment of Haruka's Badass Normal tendencies is during the Searrs invasion where she uses her brute strength to take out a pair of fully armed soldiers... and also kicking one of their tanks and somehow not breaking her foot. Wow.
While several of the characters in Baccano!! are immortal beings that have drank the elixir of life, many of the most amazing fights and other feats are performed by completely normal humans such as Claire Stanfield, Chane Laforet, Jacuzzi Splot, and Ladd Russo.
It's worth pointing out, however, that Ladd is Ax-Crazy and Claire thinks he is immortal.
In the light novels, Claire takes this trope even further - he manages to best a vampire in a fight. The author of the novels has even stated that, if such a situation were to come up, Claire would be able to win against all of the many characters in the series with little effort; the reason the author plans to avoid such a fight is simply because should Claire be pitted against the rest of the cast, the story would come to a swift and uninteresting end. What's more, at the age of 90, Claire - still a mortal - continues his eccentric activities, traveling around the world and locating treasures merely to impress Chane.
The light novels themselves note that the only character that might match up against Claire in a fight is Ronnie Schiatto. The resident demon/eldritch abomination has trouble fighting Claire Stanfield.
In Mahou Sensei Negima!!, most of the characters at the least use some kind of given magic or Ki. But some of them are at least able to touch ground with the more powerful people. Asakura showed sly cleverness — enough to manipulate anyone in the class — long before she got her spying device pactio weapon. Kuu Fei in particular was only recently able to properly utilize Ki, and hold her own against the more magically capable opponents. Although she rarely gets placed in a situation where it would become applicable due to her "clueless Muggle" status, Ayaka also proved herself capable of holding her own in battle during the school trip arc.
Negi's father also counts; even without his magic charging into an Anti-Magic valley filled with monsters is no problem if his woman is in danger.
Even without magic Evangeline can still kick Grade-A Badass Setsuna's ass all day with nothing more than some string, iron fans, and 100 years of Aikido practice.
However, saying Evangeline is normal is like saying Cthulhu is just a regular guy with really bad acne.
Rosette from the Chrono Crusade manga is the only one in her group that doesn't have any magical or spiritual energy, just some special bullets, but she's still easily the biggest badass of her group. She's defeated countless demons with powers that make them much stronger than her, and in the final volume even shoots the horn on her brother's head, even though those horns give him superhuman speed and strength and the ability to stop time. All this while a demon is draining away her soul to supply his power. In the anime she eventually undergoes Chickification, but still manages to have some badass moments.
In the anime version of Mermaid Melody Pichi Pichi Pitch, Nagisa, despite being a Dogged Nice Guy with little to no knowledge of the Masquerade, has a few moments of his own... like resisting a spell to immobilize humans, attacking the person responsible and destroying the item he was using to do it, despite being one of the only two normals in the entire cast, simply because Hanon was in pain.
The Gundam multiverse is a difficult issue, as it depends on the universe, what you consider a super power. UC would go with psychic Newtypes and SEED with genetically engineered Coordinators, G-verse is full of people with Charles Atlas Superpowers beating the crap out of others. Also in a mecha anime, you might consider fighting with a normal suit with or against way superior opponents a Badass Normal action. Examples (these characters also have to be directly compared to a newtype or coordinator, not just have them exist somewhere in that universe:
In a war where Newtypes are the only ones getting a hit on the enemy, Zeta Gundam's Jerk Jock Jerid Messa manages to rival Kamille for the entire show, despite his total lack of the Psychic Powers that make Kamille so dangerous, racking up a highly respectable number of kills. Sociopathic Soldier Yazan "I'm gonna violate you" Gable takes this even further, with his tactical skill and penchant for ambushes allowing him to head straight into Hero Killer territory.
Gundam ZZ has Masai, a young woman living in a desert, laying out some traps and fighting the whole Gundam team with an near ancient Gelgoog. And then there's Rakan Dhakaran, an Oldtype Neo-Zeon pilot who matches NewtypeJudau Ashta blow for blow, and manages to kill several other Newtype pilots, including the recently superempowered Mashmyre Cello.
MS IGLOO. Oliver May in the Big Rang. 'Nuff said. To give you an idea, he was an engineer with no piloting experience who got inside a patched-up MA made of a Bigro attached to a custom-built support platform, and completely stomped Fed forces all over A Baoa Qu, to the point they thought he was Char.
In the second movie series, there's Ben Barberry, who has the highest MS kill count achieved on foot, and Arleen Nazon, who's team was the only support Big Tray needed in their ground assault on the Dabudes in Operation Odessa.
UC Aces have a number of well known badass normals aside from Jerid and Yazaan, Hughes to date is canonically the oldest Gundam pilot (32 years old!), the Twin Stars of Carnage are so badass that they punched out Amuro's G3 Gundam once.
The 00 series features this a lot in many characters:
Graham Acre is the biggest of them, being the first character in the series to come close to taking down a Gundam in a straight fight. Had his fight with the Eins continued, he would have chopped it to pieces with the beam saber he jacked from the same unit.
Ali Al-Saachez qualifies for the first part of the first season because he was able to at least put up a legitimate fight against Setsuna (but he loses due to Exia's sheer physical power). Subverted later on when he steals the Zwei and upgrades it into the Arche. After that he just becomes another Gundam Pilot.
Sergei Smirnoff, a middle aged man in a Sino-Russian mass production suit (which is just as crappy as that premise sounds) that nearly beats the main character in his Gundam, during his specialty, melee combat. How does he do this? By exploiting the impressive strength of his suit and nearly ripping the Gundams head off. The same pilot in the same suit just curbstomped the ace of the AEU in their Super Prototype.
Cagalli Yula Atha of Gundam SEED and Gundam SEED Destiny. She's a Natural who repeatedly take on Coordinators (genetically altered humans) in combat and gets out alive. At one point she actually goes so far as to attack a mobile suit while armed with an RPG and a jeep. Ace PilotsMuLa Flaga and RauLe Creuset may also count, as neither has enhanced combat skills, and they both regularly fly grunt mechs against Gundams and come out on top. When all three finally do acquire Bigger Sticks, slaughter ensues.
Mu, Rau, and later in Destiny and X Astray, Ray and Prayer, are an odd case. In Comparison to the Coordinators, they're certainly badass normal, but they have abilities that most normals and even some Coordinators don't - a Newtype-esque psychic link, and the ability to pilot machines with drone guns like Rau's Providence and Mu's Mobius Zero, that require a hightened sense of spatial awareness to master.
Gundam SEED Destiny's spinoff, Stargazer introduces Sven Cal Bayan, a Natural pilot and elite member of the Blue Cosmos' Phantom Pain unit. Having undergone Training from Hell since a young age, Sven is able to perform as well as his own side's Extended and downs numerous Coordinators in a war where most Naturals are dropping like flies.
Gundam AGE continues the tradition with both Woolf Enneacle and Asem Asuno. Both are not X-Rounders (the local Newtype equivalent), and both can kick ass all over the battlefield regardless, using nothing but pure piloting skill, even against X-Rounders. Thanks to some Gratuitous English of Woolf's, fans refer to them as "Super Pilots".
Geronimo from Kinnikuman. In his first fight, he's just an ordinary person, but manages to not only survive getting crushed and stabbed in the torso, but destroy his opponent by screaming at him, only stopping to make his stopped heart start beating again by reaching into his own chest and massaging it. The fact that he's not even technically a superhero at this point makes this even more amazing. After becoming a superhero, though, he never wins another fight. Apparently, super powers make you weak.
Jillas qualifies as well: he has no magic ability nor wields a special weapon, but with his bombs he sometimes manages to be quite badass.
Prince Phillionel, Amelia's ogrish father, is equally as worthy, as all he has to rely on his is hulk-like strength, which is powerful enough to defeat a lesser demon and harm a low-ranked Mazoku/Monster.
While it's emphasized that usually only the characters smart enough to wear the Hunter suits are Genre Savvy enough to survive a game of Gantz, one of the facts that make Kurono such an utter Badass is his ability to not only survive playing without his suit (not by choice, he simply doesn't have access to it) but to thrive, regularly scoring more points than players wearing suits. That, combined with the time he killed a gang of vampire assassins with a homemade "sun gun", marks him out as the series ultimate Badass Normal.
Kaze technically qualities as well, having killed a few raptors with his bare hands before he gets to the point of wearing his suit. Like Kurono, the act of donning the aforementioned Hunter suits puts him well above the average Gantz participant.
Gene Starwind of Outlaw Star can be seen as a Badass Normal. Certainly, he has some fancy tech and a space ship, but when his crew are made up of a ninja-esque assassin (capable of killing anything with a wooden sword), an alien catgirl who can turn into a beast form and is nigh-invincible already and his navigator is a bio-robot, it certainly makes the plain outlaw Gene look underwhelming.
Furthering this is the fact that the most common enemies Gene runs into are Chinese pirates who use tao magic on a regular basis, while Gene has none of that (aside from his Castor gun, which is magic in an increasingly-rare gun form).
Jim, who is a 13 year old kid (Gene himself only being about 17 in the series) whose only power is being brilliant.
Shimi of the Anten Seven is a grand example. Unlike his comrades, he doesn't use any special powers in combat but is no less deadly because of it. Before his showdown with Gene, he first fights Jim, Aisha, and Suzuka all at once using nothing but conventional weapons and beats all three of them within the span of a minute. He then proceeds to give Gene a thorough trouncing and probably would have killed him had he not decided that the only honorable way to finish his quarry was through a quick-draw shoot out. He lost this showdown due to a faulty pistol, but even being shot and then buried didn't stop him from surviving and digging himself out of his grave. Truly he is a manly man among normal men.
You forgot to mention that he threw the fight by deliberately using a faulty pistol. And depending on when he decided to lose, he may have taken down Aisha and Suzuka just to make his "death" more convincing!
Many people consider Kyon from Haruhi Suzumiya already in this trope (if you include the novels, that is), as he still kicks everyone's ass without having any power at all. In the 4th novel, he courageously pwns the Data Overmind, threatening to use Haruhi to recreate the universe, should anything happen to Yuki Nagato. Badass Normal, yeah.
Worth pointing out that it's been made more or less clear that the S.O.S. Brigade members actually seem to be more loyal to him than Haruhi. Particularly in the case of Yuki who seems to take commands from him more often than Haruhi. Inspiring that kind of loyalty is indeed all kinds of badass.
Li Meiling from Card Captor Sakura, who has no magical powers at all, but still helps to take down the cards, even beating one in a straight up fight. Tomoyo would count, but she always stays out of the fights.
To fully get the picture on Meiling's Badass Normalness: Sakura is a Magical Girl. Syaoran is a pint-sized Bruce Lee Clone. Kerberos, Yue and Rubymoon are magic guardians. Meiling can't even see the Clow Cards properly... but she's still a Hot-BloodedCute Bruiser, and once she's even vital in capturing The Twins card: whereas Sakura fails to synch her attacks with Syaoran's own, Meiling does it perfectly.
Kallen Kouzuki from Code Geass is one of the few major characters without a Geass or some sort of inhuman skill, and yet she's arguably the best pilot in the series. She's able to pilot the Guren S.E.I.T.E.N. after its creator stated that even Suzaku shouldn't be able to, then narrowly defeats Suzaku in his own custom built Lancelot Albion, despite him using his "live on" command through all of their fight.
Suzaku himself could also qualify at first, even though it verges on Charles Atlas Superpower at times (like kicking auto-tracking machine guns off ceilings).
By following the above example, Lulu would have counted...at least until halfway through the first episode.
The Obstinate Arrow task force from Scrapped Princess. Centuries ago, the Guardian Factor was seeded in human DNA to produce people with the strength and will to protect the Providence Breaker when she is born. None of these people are in the task force which confuses Zefiris when they come to the Pacifica's rescue and join her side.
Demon slayer Sango is InuYasha's resident Badass Normal. Of the main Five-Man Band, she is the only one that doesn't have either demonic or spiritual power, instead relying purely on her enormous boomerang, extensive combat training, and nigh-encyclopedic knowledge about all manner of demons and their weakness. Sango is, notably, the second strongest combatant in the group, formidable enough to impress even Inu-Yasha himself. Miroku sums it up when he's forced to try to take her down while she's mind controlled - he pulls it off eventually, but not without incurring some nasty injuries, and she's clearly not fighting at full potential.
Miroku: You're truly a fearsome opponent... I'm just glad you're usually on our side!
Even Sailor Moon has a few of these, particularly Kino Makoto/Sailor Jupiter, who established her badass credentials in her debut episode by smacking around a bunch of thugs who were harassing Usagi, then doing it again later in the episode to Zoicite. This was before she awakened as Sailor Jupiter, and for that matter it was before Sailor Moon even knew she was an unawakened Senshi. When she did awaken and turned into an Empowered Badass Normal, she was one of the only Senshi to ever kill a Monster of the Weekwithout the use of Sailor Moon's Finishing Move.
Actually, Sailor Mars destroyed one in her debut episode. Still impressive, since Sailor Mars is a Miko and thus, have her own supernatural powers even without transforming. Sailor Jupiter has only her own strength, and its more than enough.
Mamoru Chiba/Tuxedo Mask. He doesn't have any of the finishing attacks of the Sailor Senshi but is usually the last one to go down, making great use of his tactical mind, physical strength (which is a little above average, but not that much), and a few key weapons. This guy is Japanese Batman.
Hubb Lebowski of Wolf's Rain starts the series as an ineffectual wuss with allergies, but by the time the world ends he's evolved into a badass that keeps up with the main characters out of sheer bloody-minded determination even after his wife dies. His refusal to die results in one of the most cringe-inducing parts of the series, since he lets himself fall off the side of what looks like the tallest mountain of the world and still lives long enough after the impact to try and smoke a cigar.
Psycho Busters has Kakeru, who, despite being the only one without psychic powers, is still able to kick ass by analyzing the flaws of each psychic and using it to his advantage. However, he may be heading towards Empowered Badass Normal instead since he does have psychic powers sealed inside him, but he's unaware of it.
Claymore's Raki, post time-skip. Unlike the ladies, he has no Youki to increase his strength, no ability to sense demons beyond the regular five senses, and he certainly is a lot less durable... He presumably heals at the same rate as a normal human. He's just a regular guy with seven years of sword training under his belt... And he easily kills the sort of youma that was giving Clare trouble at the beginning of the series.
Captain Bravo from Busou Renkin would seem to fit this trope. Although he does have a kakugane, it is simply a metal shield, however he is able to destroy the ground and kill people with a single punch, pull out of a spinning kick in mid air, etc.
Looker (Handsome in Japan) in Pokémon. He first comes off as a slightly bizarre detective, but he dares to actually fight bad guys HIMSELF instead of relying on any Pokémon.
Bad girl Domino in Pokémon: Mewtwo Returns held off and pushed a Scyther away herself among other things. She was one of the few human characters who did not have a Pokémon and did things herself quite well; she was Crazy-Prepared.
Looker and Domino both impressed in their worlds, but it's more common in the Pokémon Special manga for trainers to fight against Pokémon on their own, usually out of necessity. You could easily expect such out of Bruno and Chuck, but then there's Sapphire, who is first seen fighting a Seviper alongside her Torchic, and later has her Aron chew off a bit of a fountain before breaking it off herself to save another Pokémon trapped inside!
And in Best Wishes both of Ash's traveling companions, Iris and Cilan prove that they can battle a Pokémon without the need of using one. Iris has demonstrated many ninja-like movements being incredibly fast compared to other Pokémon, has shown to dodge many attacks coming to her and was able to befriend Drilbur that later evolved into her powerful Excadrill because of it. As for Cilan, while he does more thinking than actual battling, he's shown to be a pro at fishing. He battled a Stunfisk just using his fishing rod and caught it after Ash caught his Palpitoad.
Ash himself, as well, if his ability to use Aura, which he never uses anyway is ignored. With his personality and young age, it's very easy to forget that he's faster and more agile than a lot of his Pokemon and has a tendency to jump into the action himself when things start getting rough and come out little the worse for wear.
Jubei of Ninja Scroll, facing various mutants and mystical opponents with nothing but a sword and his skill with it.
Kodachi Kuno from Ranma ˝, who is considered one of the most dangerous characters, despite being arguably the weakest character of the semi-regular cast as it comes to martial skill (even Akane is regularly able to divert her direct attacks). She combines levels of Manipulative Bastard, Fighting Dirty and Crazy-Prepared. Often she makes use of her pet alligator and fellow St. Hebereke Gymnasts to wear down Ranma and others and keep them running around in her plots. Usually, she gets out of her storylines with little to no damage to herself and Ranma and Akane feel lucky to be alive.
Otoha of Karas is an interesting example, in that he is a Badass Normal compared to himself. He was originally a yakuza enforcer who liked killing gun-toting mooks with nothing but his sword, and was actually able to hold his own (if only barely) against a full-powered mikura. Then he turned into Karas, and anything he fought just kind of died.
Hei from Darker Than Black was a Badass Normal until the end of Heaven's War, when he ended up with his sister's Contractor powers. As a teenager who had nothing more to go on than ninjutsu, he still managed to earn the nickname "The Black Reaper" from taking down so many Contractors.
Elfen Lied has soldiers that are mostly mooks that stand no chance against the super powered diclonius unless in large numbers; except for Bando who is a Badass Normal that puts up a good fight whenever he can get the chance to. Despite being augmented with high-tech prosthetics, these are actually still handicaps for him. He qualifies by being Crazy-Prepared in that he will only face her in an open space on the beach that he keeps clean of debris for months by patrolling it daily. He is also very ruthless and skilled.
Several other characters display this trope. The Agent defeats several Diiclonii including Lucy. The Unknown Man manages to defeat Nana and get the better of Bando temporarily. Even Kurama charged Lucy head on without realizing that she had been Brought Down to Normal.
Minamoto from Zettai Karen Children. The protagonist of the series, he's the head of a level 7 three-person esper(ESP user) team. He has to go up against freakishly strong espers every mission. Some of the things the espers can do is levitate fire trucks like they were nothing, ward off bullets, and read/screw up your mind. Despite this, he manages to constantly even the odds with superior tactics, usage of technology, and plain old kindness and understanding. No wonder he's respected by espers on both sides.
Domyouji from Kurogane no Linebarrel is a common human being who can fight toe-to-toe with nanomachines-enhanced individuals (who can easily punch normal people unconscious in a single blow, and have repeatedly done that).
In Soul Eater, Mifune is one of the few characters who is neither a witch, a magical weapon, or the wielder of a magical weapon. He's just really really impossibly good at swordfighting.
Bardroy of Black Butler is the most normal out of the whole house staff. He isn't a super competent badass like Sebastian, doesn't have super strength like Finny, and isn't a ridiculously skilled sniper like Mey-Rin. He gets by on his apparent strategical know-how and skill with a rifle.
Bunshichi Tawara from Tenjho Tenge stars in a series filled with formally trained martial artists, people with inherited super powers and the mortal incarnations of Amaterasu, Tsukuyomi, & Susanoo, yet still manages to be (arguably) the strongest fighter in the series without any kind of training or supernatural enhancement. Needless to say this has given him quite the reputation in series.
Mifune . While the other leaders of the Shinobi Alliance can do stuff such as creating a sea of sand, obliterating mass, healing with such speed that it equals immortality, manipulating lava and acid alike, and dramatically augmenting one's physical abilities with lightning, Mifune just draws his katana fast enough to prevent even high-level ninja from performing the hand-seals needed to use their special techniques.
Fans of Voltron would know that Sven, the original Blue Lion pilot, is one. He survives being nailed in personal combat by a robeast, and survived a plunge from the heights of Castle Doom. Even after that, he was in good enough condition to fly Blue Lion again.
Makoto's father in Sacred Seven is a (sorta) ordinary butler in a series populated by extremely powerful monsters called Dark Stones. Any character who has ever fought one needed special powers of their own, a Mini-Mecha, or a big lasergun. Yet, he managed to hold his own against one, however briefly, with only his barehands and two Hidden Weapons. He later managed to defeat it by blowing himself up.
In the manga Medaka Box, Zenkichi Hitoyoshi, one of the normal main characters, is able to hold own his against an Abnormal(one who has a special ability caused by their personality) and a Minus(similar to an abnormal but a minus is morally corrupt).
He is possibly even an inversion to the trope, having received a power that nullifies fate in battle, meaning that he will receive no second chance, nor a Deus Ex Machina, nor any other sort of reprieve from his situation.
Yahiko Myojin from Rurouni Kenshin. While his idol, Kenshin is a master of a Sword Style that grants both Super Speed and Super Strength to the user and is a Shell-Shocked Veteran, and Sanosuke is a bullet timer brawler whose skin is so tough he can shrug off a direct stab from a normal human being with no damage, Yahiko is a 10/11 year old boy who took down a Juppongatana by riding the blast wave of his own bomb and smashing him with his shinai!
Mizushima Toru, protagonist of Iris Zero. Interestingly, he lives in a setting where amongst his age group, having eye powers is the norm. His lack of iris earns him the titular nickname. He manages to get by through Awesomeness by Analysis, and utilizing his friends abilities more skillfully.
Genjuro Kazanari of Senki Zesshou Symphogear is just a regular dude. He eats, sleeps, and watches movies...and he's easily capable of handing the uber powerful Gear users their asses, without breaking a sweat.
Heivia and Quensar who manage to destroy fifty foot tall robots with just some heavy explosives, some weak guns and their wits.
Mr. Satan / Hercule in Dragon Ball Z. All the showboating aside, he is the best non-superpowered fighter on the planet and won the World Martial Arts tournament fair and square (since none of the superhumans competed that year). His strength is nothing compared to the Z Fighters and their foes, but no other human could match him...except for his daughter Videl, who not only surpasses him in strength but learned the Z Fighters' flying technique from Gohan.
It should also be noted that he managed to persuade the series' final Big Bad, an Eldritch Abomination that Goku had trouble with when he was going all-out, to do a Heel-Face Turn. And when the evil side of that character manifested itself, he played a crucial role in destroying it. There's also the fact that he is the only human—strike that, the only protagonist, to never die in a series where Anyone Can Die.
Viral from Tengen Toppa Gurren Lagann is a Beastman, and thus does not have the ability to use the all-important Spiral Power no matter how much he may want to. Still, he manages to take Kamina's place as the pilot of the Gurren-half of the titular mecha in the latter half of the series. Despite have less spiral energy than any human, he's picked for the job and holds his own because he's incredibly skilled with Gunmen-piloting to the point where he doesn't NEED Spiral Power.
The same, shockingly, applies to none other than Kamina!Word of God states he has very little access to Spiral Energy, yet is capable to fight toe to toe with dangerous adversaries because he's able to draw energy from Simon's endless pool of Spiral Energy. Arguably, this makes him even more awesome than he already was!
Ryou from Popcorn Avatar is still able to handle a fight against the gods on his own, despite being no longer affiliated with an Asura.
Essentially the entire point of Battler from Umineko no Naku Koro Ni. Not only does he refuse to believe in magic, his disbelief in magic makes him immune to it.
Hayato Kawajiri in Jojos Bizarre Adventure is only a grade school kid with no Stand ability, but still manages to root out Kira, the Big Bad of part 4, after he kills his father and steals his identity before anyone else. Even after said Big Bad turns him into a living bomb that kills anyone who even questions him about Kira's identity before starting a "Groundhog Day" Loop, he still resolves to put a stop to it even if he has to kill him himself. Eventually, he manages to Out-Gambit one of the most Dangerously Genre Savvy villains in the series, and for an encore Jumps On A Proverbial Grenade to take one of Kira's bombs to save Josuke and Okuyasu, only managing to survive because Josuke managed to act fast enough with Crazy Diamond to save him. Again, this is a grade school kid.
Hidetora Tojo from Beelzebub manages to fight Oga on equal ground despite the latter being demonically powered and later manages to beat 100 demons by himself.