Cheradenine Zakalwe from Iain M. Banks's Use of Weapons epitomises this trope. He gets a little less cocky about it after being decapitated, though.
"Gentleman" Johnny Marcone in Jim Butcher's The Dresden Files. Managed to bring Chicago's organized crime under his (relatively organized) reign, and has gone to toe-to-toe with the things that go bump in the night on multiple occasions. White Night ends with him becoming a recognized body in the supernatural community, able to defend his rights if any signatories to the Unseelie Accords infringe on them. There are twenty such bodies throughout the world, and Marcone's the only mortal. His biggest moment has to be when he was hung upside down and managed to kill a werewolf with a throwing knife. In the dark.
Karrin Murphy. She's faced down rampaging werewolves, vampiric minions, and an army of ghouls, despite being a squishy mortal. It does help that she's an experienced cop and damn good at Aikido, though.
She's also described as looking like "someone's favorite aunt" and a cheerleader, with a button nose, blue eyes, and blonde hair. Five feet and a hundred pounds of Bad Ass.
However, if she's wielding one of the Holy Swords (which she only does a last resort...for now), she moves up into the Empowered Badass Normal category.
Charity Carpenter was once a practitioner of magic, but gave it up after Knight of the Cross Michael Carpenter rescued and married her. In Proven Guilty she takes on the Badass Normal (and Mama Bear) role in the team assembled to rescue her daughter Molly from Arctis Tor. She uses no magic, only the muscles and skills she has built up as her husband's armorer and sparring partner.
Must run in the Carpenter family. The latest book, Ghost Story, has Daniel Carpenter, Charity and Michael's oldest son, who holds his own in a knife-fight with a magically-powered adversary on the basis of the training from his parents.
Marcone's bodyguard Hendricks also qualifies, managing to come through fights with super-ghouls and Denarians unscathed.
Kincaid claims to be as human as the protagonist, and spends his first appearance in Death Masks taking down multiple Red Court vampires with a golf bag filled with disposable shotguns with Dragon's Breath rounds. In his next appearance he's revealed to be several hundred years old and a definite supernatural of some kind, though exactly what kind is unknown.
Kyja of " Farworld" is the only person in her entire world who can't do magic. she learns to kick ass with a sword, and when the other hero shows up, he can't do ANYTHING because he can't control his magic. the first book is mostly kyja saving his sorry and crippled butt from baddies.
Tavi in Jim Butcher's other series, Codex Alera. He is uniquely powerless in a world in which all humans can control elemental spirits called Furies, though he compensates for this near-crippling deficiency by becoming a courageous, insightful, empathetic, and generally badass leader. This may be a slight exaggeration of this trope, as while he is normal by our standards, in his world he is considered a (sub-normal) freak. By the end of the third book, it is explicitly stated in-story that he is far more capable than most people with great Fury-powers. And then he becomes a Empowered Badass Normal with the full furycraft ability of the First Lord.
In the Young Wizards series, Kit's sister Carmela is definitely badass normal when she manages to not only make an intergalactic voyage, chat with Aliens, learn The Speech, and wound the series antagonist in his plan of the week in Wizards at War, but she does so without any skill in wizardry at all, just by hanging out with Kit and Juanita.
In the same series, Juanita's mother sends the series antagonist back despite having no wizardry powers... all she just does is because it's inside her mind, give him 'all the pain she experienced in her life. This includes childbirth. Too bad you didn't get to see her pwn him again...
Geoffrey Spazmo from Ben Elton's Gridlock. Spastic, but smart and brave as a lion. Kills two professional assassins with a coffee maker and a bottle opener. Still dies in the end... sniffle...
Superhuman the book by Michael Carroll (the author of the The New Heroes trilogy), takes place in a world where not everyone has superpowers, but people with powers, both villains and heroes alike, aren't a new occurrence. The book details the efforts of four kids' efforts to stop a four thousand-year plot to resurrect and evil king, Out of the four kids there are two girls and two boys. Three of the kids have amazing superpowers. The first child to be introduced in the book, however, is the only one without powers. This boy, Lance, is presumably the youngest in the group, being only 14-years-old. However, his manipulative skills and way with words made him boarder on Magnificent Bastard status. Even those that didn't like him at first admitted that they needed his help off the battlefield.
Steven Erikson's Malazan Book of the Fallen has a number of these, with the assassin/warrior Kalam Mekhar being perhaps the most obvious.
In the Conclave of Shadows trilogy, orphan Talwin Hawkins was trained in a wide variety of skills to become a medieval fantasy version of James Bond. He managed to kill his sworn enemy in a public sword-fighting tournament in a legitimate way, also winning the title of the best swordsman in the world. Which is a gambit to get himself hired by one of the most cunning ruler in the world, passed his magical lie-detectors, got his hands cut off and magically regrown, and killed a powerful necromancer with only his wits, sword, and a well-aimed steel ball.
Waylander from David Gemmell's Drenai books is the epitome of this trope. Despite being rather mediocre with a sword, his nerve, accuracy and ruthlessness (and a nifty double crossbow thing) mean he is the world's foremost assassin. In the later books he becomes known as "The Gray Man" (apparently one of the embodiments of death) and leads the resistance against a bloody demonic invasion despite being an octogenarian.
Arguably three of the other recurring characters of the series (Druss the Slayer, Skilgannon the Damned and Talisman/Ulric) would easily qualify, as each one of them can only be described as a force of nature incarnate but as they, individually, at some point have received magical weapons/enchantments which have made them stronger than before.
Although, in the cases of Druss and Skilgannon, they were massively badass both before the magical weapons and after they had put aside their weapons (Skilgannon), or taken a terrifying jaunt through hell that rid their weapon of its power (Druss).
In Walter B Gibson's novels about the Shadow, the Shadow relied on the techniques of Houdini (which Gibson had gotten the permission to use and write about from the estate of same) to rescue himself.
Chase from Terry Goodkind's Sword of Truth series has no magical ability, but can outfight anybody on the planet, and in the beginning of the series has a day job that involves him fighting demonic hounds that hunt by the sound of human hearts. He's mentioned as being "far tougher than he has any right to be", and carries enough weapons to equip a small army. Not surprisingly, his adopted daughter becomes a Little Miss Badass later in the series.
Roran from Inheritance Cycle. In a world where elves have the strength of ten men, dragons grow to the size of large hills, and magicians tear castles apart with their minds, Roran is armed with only his determination and a metal hammer. When his village becomes condemned by The Empire, he uses his potent charisma to convince his people to flee their homes and travel from the northern tip of Alagaesia to the sun-drenched country of Surda in the far south, avoiding Galbatorix's troops all the while. He joins the Varden, kills the Twins (two extremely powerful magicians) with his hammer, kills 193 men in one go, survives being given 50 lashes to the back by Nasuada for insubordination and is up and fighting again a few days later, journeys across Alagaesia to rescue his kidnapped girlfriend from the mountain lair Helgrind, wrestles a battle-crazed urgal to the ground until the beast surrenders and acknowledges Roran as the stronger, and rises his way up to a commander in the Varden after only a couple of months of service. And he does this all without any magic whatsoever. Yeah, Eragon doesn't look so impressive next to that, does he?
Lampshaded by Katrina's, Roran's love who he marries in Brisingr, who remarks that the feats Roran accomplished without magic require greater courage than anyone else in Alagaesia has.
In Song of the Lioness, Alanna meets a fighter named Liam Ironarm, who doesn't have magic and actively shuns it (because he's afraid of it). He's the Shang Dragon, and one of the prerequisites for Shangs is that they be un-Gifted.
Protector of the Small's protagonist, Keladry of Mindelan, is the only one of Tamora Pierce's protagonists without magic of any kind. This was a specific effort to make her and her successes more relatable to young readers.
Tiphaine (and her late partner Katrina) from S.M. Stirling's Emberverse series. They're ninja-style spies and assassins, but portrayed as being normal humans whose only abilities come from intensive training. Occasionally they skirt the edge of Charles Atlas Superpower, but never quite slip over.
Alexander Storm from the Deathstalker series. He was one of the few characters not either a cyborg, super-gladiator in disguise, or afflicted by an ancient alien maze, and yet he still managed to survive half the fights that hit him, right up until his Face-Heel Turn.
Edward from the Anita Blake series has no superpowers whatsoever and regularly goes up against vampires and shapeshifters for money and at one point against a creature that turns people into pulp for no apparent reason. Similarly, the members of RPIT, most notably Dolph and Zerbrowski, are at least acknowledged and often feared by members of the supernatural community.
Anita is a Federal Marshall, necromancer, succubus, licensed Vampire Slayer, and part of a triumvirate with the master vampire of St. Louis and leader of the local werewolf pack, plus the mess of lycan blood she has in her. Edward...is a guy in his early 30s. Monsters call Anita The Executioner, they call Edward Death.
Earth-born Hari Michaelson in Matthew Stover's The Acts of Caine is, in his Caine persona, a superstar in a D&D-inspired parallel universe. He goes up against the most brutal warriors of a fighting species, thaumaturgists, humans channelling the powers of their gods, and, well, gods. He beats them all through a combination of training, a few semi-magickal meditative disciplines, his knives, and, when he's physically incapable of anything else, extraordinary tactical ability and a healthy dose of badassery.
In book three, Stephenie Cord also becomes a badass normal.
Dioxin also becomes a badass normal after he loses his powers.
Rod Gallowglass (a.k.a. Rodney [too many middle names to list] D'Armand) is the Badass Normal in Christopher Stasheff's Warlock of Gramarye books, or at least until around the end of the third book.In a World where everyone is at the very least a latent telepath, a few thousand people are full-blown, teleporting, levitating, telekenetic, you-name-it-it's-there telepaths. All he has is an advanced knowledge of Science and History, an epileptic robotic horse, training in all types of weapons, and an 8th degree black belt. He still manages to fulfill his role as The Chessmaster in guiding the planet towards democracy, plus raise the four most powerful telepaths to be born... ever. Even after he discovers his own "witch-powers" he prefers to rely on the skills he learned beforehand.
Later in the series, Yorick becomes the Badass Normal. Specifically in The Warlock Is Wandering. Isn't it amazing how well a Neanderthal, who can't manipulate symbols due to a lack of prefrontal lobes, can manipulate the rules and people around him?
Even later, in the semi-spin-off Wizard series, Magnus Gallowglass takes on a partner, Dirk Dulaine, who is decidedly a Badass Normal. You don't get much more "normal" than Dirk's genetics. His entire ancestry comes from maybe 100 individual people, who were then cloned. A few centuries later, and the servant population bears a striking resemblance to each other.
The Seekers of Truth have a couple. Trigger uses technology and an eagle eye, and Grizzly has lived in the woods for the last 25 years honing his body and his instincts, until he can read the sounds of every animal in his woods.
As explained on the relevant page, the pilots of Rogue and Wraith squadrons. In a galaxy where Jedi and trained commandos struggle, pilots still hold their own. Wedge Antilles might not be as physics-defying as a Force-Sensitive pilot, but he's counted with them as the best pilot alive, and is at the very least notable for being the only person to survive both attacks on a Death Star.
He is Blackmoon Eleven. The greatest pilot of all time. And when he led a battle that was supposed to be a stalling tactic, he unintentionally won, even though that was exactly what they were trying to avoid. It was lampshaded immediately afterwards by Tycho.
Samwise Gamgee. A gardener at the beginning of the book. By the end of the trilogy... well, I know it's the film, but Shelob's lair is his CMOA. I mean, seriously. Sauron and Morgoth had trouble containing Shelob and Ungoliant, and Morgoth was the most powerful of the Ainur and Sauron the most powerful of the Maiar, but Sam goes and punches out Cthulhu.
It's even more awesome in the books; in the film, he and Frodo just use the Phial of Galadriel as a torch during his battle, but in the books, Sam actually uses it as a weapon. His willpower alone makes it burn so bright that Shelob's eyes are damaged, causing her to retreat.
In the books, Sam's badassness is really only visible in Shelob's mind. In the film, where it's harder to show that sort of thing, depicting the physical actions only would have made it look like any schlub with a good flashlight could have done the same thing. However, a short time later Sam goes into a fortress filled with orcs. Alone. He comes back out again, too... with the guy he went in there to rescue. Definitely badass.
The orcs who find Frodo's body after the battle know that there's been a SERIOUS badass through there. They think it's some ancient Elf warrior with a huge sword and probably an axe, too. The fact that it's a halfling who until a few months before hadn't thought of an axe as anything but a tool to cut wood with and had probably never SEEN a sword only makes it all the more badass.
In an earlier encounter with Shelob Frodo also applies (he also has other badass moments throughout the book, such as the wight encounter, the defiance of the Riders at the Ford, and the attack on the troll in Moria). Neither he nor Sam can see what is chasing them, and they are both running away, when Frodo decides that running is useless, turns round and goes to face whatever it is, and ends up using his will to subdue Shelob. Pity that all of Frodo's badassness was cut in transition from print.
When Sam and Frodo are trapped in the fortress of Cirith Ungol by the supernatural Watchers, they use the Phial of Galadriel as a focus and break the wills of the guardians, long enough to get through! The biggest badassery of it all, Frodo carries the One Ring to the very lip of the Cracks of Doom, in the Chambers of Fire where the power of Sauron is strongest in all the world. Only there, only there, weakened by over a year of worry, hunger, thirst, torture, labor, travel, and the cumulative burden of the Ring, only there where Sauron and the One Ring are strongest in all the world, is Frodo Baggins finally overcome by the Ring. Frodo does ultimately fail in his quest, and has to be rescued by the Mercy of God, acting through Sméagol, but Tolkien has said specifically that nobody else in the world east of the Sea, nobody, would have gotten nearly as far as Frodo did.
On the subject of the worlds of Tolkien, Beren son of Barahir. A Man, mortal born and mortal doomed, who fought with the last band of guerilla warriors led by his father after Morgoth's forces overran his lands, last survivor of the ambush the slew them, and then he waged a one man war that was so successful that eventually Morgoth set a price on his head equal to the High King of the Noldor. Even then, most of his forces would flee if the rumor had him in area rather than try to capture or kill him. So Morgoth finally sent an army to track him down, with balrogs. Think about that, he had become enough of a nuisance to rate that. Finally forced to flee from that magnitude of a threat, Beren became the first Man to enter Doriath, successfully penetrating Queen Melian's defenses that had held off Morgoth's forces for centuries. Then, in order to win the hand of his lady love, he is successively tortured by werewolves, defeats two sons of Fëanor in battle with them mounted and him on foot, narrowly escapes death repeatedly, with his lady he penetrates Thangorodrim itself to take a Silmaril from the crown of the greatest evil power of all time, then he is slain and becomes one of the only Men ever to return to physical life (briefly) after his physical death. As an afterthought, he leads an army of Ents to overcome an army of Dwarves after a massive charlie foxtrot incident that led to the destruction of Doriath. Beren was, very probably, the greatest badass in the history of Middle-Earth other than some of the Ainur.
Repairman Jack goes against things straight out of Lovecraft's worst nightmare with little more than healthy dose of righteous anger, resourcefulness and enough firepower to singlehandedly wage war on a small country. The results aren't pretty but awesome all the same.
A slightly skewed version is Bahzell Bahnakson in David Weber's The War Gods series. Among his own people, Bahzell is a Badass Normal at the beginning of the books. He is chosen by the God of War to be a Champion because of this.
Roland in The Dark Tower. The whole Ka-tet become Badass Normal by the end of the book including Oy, the Team Pet.
In BIONICLE, Vezon, although never actually fighting, manages to remain a main character, not dead, unbelievably unscathed after being captured by the worst torture master in the MU (with the building collapsing), and unmutated by Pit Mutagen, and all without powers! And plus he's got practically no mind, so he might even be considered handicapped...
And Mazeka, minus the handicap. He's a Matoran (the verse equivalent of Hobbits) who learned how to "fight clean, fight dirty" and kill pretty much anything.
Speaking of the Pit, Hydraxon has no elemental powers at all. He doesn't even have a Kanohi Mask. What does he have? Slightly enhanced hearing, some throwing knives and A LOT OF GUNS. It took an earthquake from a 40 million foot tall robot crashing into a planet to kill the first one. The second one was just as tough. He's the only thing that the former Pit prisoners are afraid of and he trained the Toa Nuva.
In the Wild Cards novels, a character who was never infected with the Wild Card virus is called a "nat" (short for natural). In the first book, a nat named Yeoman managed to take down a teleporting Ace.
In the Discworld novels, Commander Vimes has fought a dragon, outwitted a vampire, outran a pack of werewolves and is the only person to have bested the Summoning Dark, which is an ancient evil that works by corrupting and controlling men's minds. He does all this even though he is a middle-aged man with no powers whatsoever, going as far as to refuse magical help from wizards.
Vimes has survived 9 assassination attempts by the Assassin's Guild without any injuries. After that, they gave up and started using him to test (read: punish) their trainees.
Rincewind is a 'wizzard' with absolutely no powers and an inability to learn any spells. Even so, he is able to get out of just about any situation with a mixture of fear and speed. One time he manged to survive the disappearance of a dragon he was riding by punching into an alternate reality through nothing but a fervent desire to remain airborne.
Despite living on the planet Petaybee, where most people have adapted to perform well in very low temperatures, Yana Maddock, who was born off-world, still stands out as one of the most badass characters in the series.
Niko Leandros from Rob Thurman's Cal Leandros series has a vampire love interest, a half-human, half evil-fae brother, and is friends with Robin Goodfellow. He has weapon skills worthy of Ghengis Khan and Alexander the Great according to his brother and he's one of the only humans respected by the supernatural world in New York City. He also had Heroic BSOD when he thought his brother was dead and killed 5 Ccoa, 15 Cadejo, and a Gualichu single handedly with only his katana.
Robles demonstrated this by having Lucretia take apart a combat drone. It took the superhuman but untrained Lucretia several minutes to take out that one alone. Robles then took out ten of them with only her bare hands in roughly 10-30 seconds without even being tagged.
Kali demonstrated this by just short of killing Lucretia.
To some degree, all of Lucretia's guard detail and the bank robbers she interferes with are this.
In the Xanth books there's Bink. In a land where everyone has a magic talent (unless they've mutated further away from human and are inherantly magical, like Chameleon) and man-eating trees (several varieties...), dragons... let's just say that Xanth in the early books is a Death World and leave it at that, Bink has no magical power whatsoever and has been forced to become a Badass Normal because of this. Subverted when it's revealed that Bink's talent is that he can't be harmed by magic: it manipulates all magic around him into acting in his best interests, combining Plot Armor and Magnificent Bastard, and by concealing itself it protected Bink by making him appear harmless and forcing him to become a Badass Normal. Double Subverted at the end of the second book, when an Eldritch Abomination reverses his talent, causing it to manipulate all magic around Bink in order to harm him and Bink is able to survive due to his skills.
Boba Fett is shown to be quite the badass in spite of lacking Jedi powers. He's extremely physically strong and able to handle a lightsaber much like a Jedi. He even fights Mace Windu to a standstill in a novel as a 14-year-old boy! As quoted by Han Solo himself: "There are six, maybe seven people in the entire galaxy who can beat Boba Fett in a fair fight... and I'm not one of them."
The Mandalorians in general are known for being Jedi killers during the many wars that dealt with Force-users.
Han Solo. Just shooting Darth Vader off Luke's tail in the first movie would have done that, let alone everything else he manages in the Expanded Universe novels...
Grand Admiral Thrawn. Who could not only command a mad Jedi Master but also cut him down to size when he got too big for his britches.
Master C'Boath: "You doubt the power of the Force, Thrawn?"
Thrawn: "Not at all. I merely present the problems you and the Force will have to solve if you continue with this course of action. For instance, do you know where the Coruscant sector fleet is based, or the number and types of ships making it up? Have you thought about how you will neutralize Coruscant's orbital battle stations and ground-based systems? Do you know who is in command of the planet's defenses at present, and how he or she is likely to deploy the available forces? Have you considered Coruscant's energy field? Do you know how best to use the strategic and tactical capabilities of an Imperial Star Destroyer?"
Vlad Taltos. He's a fairly good witch and a rudimentary sorceror, but this world has Dragaerans who live for maybe 3000 years, are stronger and taller, and at least three of the ones he hangs out with have "Great Weapons", and are very skilled at sorcery. This is before he gains Lady Teldra/Godslayer.
Conan the Barbarian does not have any magical abilities, but he regularly overcomes evil wizards and supernatural horrors with his strength, speed, toughness, and natural cunning.
Jenna Heap in Septimus Heap, despite not having Magykal powers, did knock the Toll-Man and Jakey Fry down in Queste and Syren respectively.
Last Legionary: Douglas Hill's Keill Randor does have his skeleton rebuilt with an unbreakable organic alloy...but he was a Badass who was a highly skilled fighter to start with. Man cannot kick ass on unbreakable bones alone.
Leaf from theLiterature/ Keys to the Kingdom books by Garth Nix. Poor girl gets dragged into Arthur's adventures even though she barely knows him. She continues to help him and do what needs to get done on Earth even after being kidnapped and forced to serve for months on a ship in Drowned Wednesday. She stands firm against monsters and Denizens from the House and rescues Arthur almost as often as his official sidekick Suzy Blue does.
In the web-novel Domina, Adam Anders doesn't have a power, and can't use the toy maker. He compensates with lots of guns.
Laura Webster, protagonist of Islands in the Net, is a PR agent and mother of a young child who, without any warning, finds herself up against drug dealers, computer hackers, super-soldiers, and worse. She is also tough, fast, smart, and determined enough not to let any of them stop her.
Richard from The Power of Five has moments where he's at least as competent as any of the Five. Oh, and then there's Lohan in the final book, who casually (and efficiently) murders his way across South America in his quest to survive and help Matt.
After The Golden Age has the Hawk, a Non-Powered Costumed Hero who is famous as the only vigilante superhero in Commerce City not to have any actual superpowers. He is nonetheless one of the city's most skilled and respected crime-fighters.
The protagonist from Legacy The Tale Of The American Eagle is the American Eagle, who is an accomplished martial artist, accurate with throwing knives, speaks various languages and practices sports like le parkour and BASE jumping in his day to day operations. In an effort to add realism, however, he is shown to be horribly wounded after each 'act' of the novel, and is often healed by extranormal means to get back in action quickly.
Please Don't Tell My Parents I'm A Supervillain: A number. Cybermancer and Penny's mom who went by The Audit in her heroing days. Probably Miss A, though her powers are never fully explained, if she has any. Witch Hunter seems to be one as well.
In Rogue Sorcerer, Jaren is able to hold his own against Aiden in a fair fight, and also protects his village from roving bandits on his own. Lyr also certainly counts, easily overpowering Aiden even after pressing his Berserk Button.
In Spectral Shadows there's the Amazing Detective Agency and Ratzo DiCaro qualify; heck pretty most of Noir would qualify, since they don't believe in using magic or powers of any kind. Also Jasper is this When he's outside of the Shadow Armor.