Rory Williams: "I have a message and a question: a message from the Doctor and a question from me. Where. Is. My. Wife?" (the Cybermen do not respond) "Oh, don't give me those blank looks. The Twelfth Cyber Legion monitors this entire quadrant. You hear everything. So you tell me what I need to know, you tell me now, and I'll be on my way.
Cyber Leader: "What is the Doctor's message?"
Rory Williams: (one of the Cybermen's space stations explodes in the window behind Rory) "Would you like me to repeat the question?"
On Buffy the Vampire Slayer Xander occasionally laments how all his friends and allies have superpowers—Slayers, witches, werewolf, watcher, ex-demon, vampires—yet he's the guy "fixing the windows". Nevertheless, he is the only one to fight the good fight for more than eight years without ever straying from the path. He once defeated the Big Badby simply talking after it became clear no one on earth was strong enough to stop her.
On Angel, Gunn is the epitome of the Badass Normal in the first 4 seasons. He's the only one who's received no special training, isn't a centuries-old demon or magically empowered in some other way. He's been fighting vampires since he was fourteen and, in the fourth-season episode "Players," he took out six Asian warrior monk-types without getting hit once. Gunn and the street-gang members he used to lead also fought vampires with more success and less casualties, using improvised melee weapons, than the US government's secret black-ops demon-fighting organization was able to manage even using chemically-augmented super-soldiers and energy weapons. Admittedly, the government wasn't just hunting, but studying...
Fred, who, in the later years of her short life, was very adept with a flamethrower (despite being a good ol' southern girl). Cordelia could also qualify as this before her demonization; she could corner Angel without much of an effort after a couple of training sessions.
Angel also gives us Holtz and Justine. Holtz has managed to capture Angelus/Angel more than once.
Noah Bennet in Heroes. He has defeated superpowered people simply by knowing how they work and going around them. He thought in Japanese to counteract telepath Matt Parkman's mind reading, arranged an escape from a prison meant to contain superpowered people, and kept an electrically-powered super under control by soaking her in water. In one of the online comics available, he defeats a man who can destroy things with his hand by grabbing him by the wrist and beating him unconscious with a baseball bat. Why not shoot him? It was inconvenient to figure out a way to go get his gun which was near his family.
In "The Butterfly Effect" (season 3 episode 2), Bennet is being held in a Level 5 cell. Level 5 is normally only for the most dangerous superpowered people, but Bennet is so badass without powers that the Company feels the need to keep him in Level 5.
During the two-parter Eclipse coupling, he became the most powerful character in the show after all the others lost their powers. Suffice to say quick wits, extensive knowledge of powered individuals and a handgun are all he needs!
And a box cutter.
Season 5 seems to be topping the sundae that is Bennet's badassness. Gut sliced up by speedster assassin? Silly Edgar, you can't kill Noah Bennet. Bennet's up and walking again by the end of the week. He gets his revenge in "Let It Bleed", locking Edgar in an industrial freezer for an indefinite amount of time, then tying him to a chair and spending several hours practicing the Jack Bauer Interrogation Technique on him.
The Bennet family is pretty Bad Ass (even Claire, but she is superpowered), considering Sandra goes with her daughter on a rescue mission to save Meredith and Lyle goes after Ted with a baseball bat and throws a bucket of water on Elle, in the style of dear old dad.
Ando, Hiro's best friend and somewhat-sidekick with absolutely no powers whatsoever. In the first 3 volumes at least. "Somewhat-sidekick" because Ando's common sense and steadfast loyalty to his geeky, Idiot Ball carrying buddy has a tendency to save the day more often than any superpower.
Volume 4 gives us Emile Danko, a veteran soldier who, while he does occasionally need some help from the likes of Noah or Sylar due to his inexperience at dealing with superpowered people, proves to be a very dangerous adversary due simply to his cunning and creativity (like framing a hero as a suicide bomber). You can call him the Anti-Bennet, due to having the same amount of badassness, but far looser moral standards.
Peter Petrelli in Season 3 when he lost his powers. He still managed to get into Pinehearst, and would have killed his Too Powerful to Live father.
Deaton, the veterinarian in Teen Wolf. COMPLETELY UNFLINCHING in the face of the Big Bad Wolf, even after getting a chair thrown at him. "We. Are. Closed."
The Argents and werewolf hunters in general make a freaking point of being this, using wolfsbane-laden arrows and bullets to take down wolves.
Many of the recurring non-supernatural characters in True Blood, but particularly Jason Stackhouse (less so in Season 1); he defeats Steve Newlin with a paintball gun, chucks Bill out of his house when he starts getting uppity, and saves Tara from an insane Yandere vampire.
Xena: Warrior Princess has Autolycus and Gabrielle. Autolycus is the King of Thieves and Gabs Took a Level in Badass from being a village girl who dreams of adventure to taking Xena's place in the series finale, with no suspect ancestry.
Xena herself wanders in and out of this trope. It's possible that she doesn't apply at all, because it's implied but never confirmed that she's Ares' daughter. The standard assumption in-universe is that she's 100% mortal, and she can kick ass and take names with the best of them but it can verge on Charles Atlas Superpower at times.
Iolaus from Hercules: The Legendary Journeys, which shares a verse with Xena, qualifies too. The man hangs out with Hercules and regularly aids him in combat, and saves his ass more than a couple times.
Supernatural gives us Dean Winchester. When the guys he stops the Apocalypse with include an angel, an archangel, a demon bigwig, a much more experienced hunter and his brother who can exorcise demons with his mind and you're STILL the most awesome, you're pretty badass.
Bobby Singer could also be included as this, especially after he ends up in a wheelchair.
While the Doctor of Doctor Who is not quite normal, many of his companions fall into this category, particularly Leela, Jamie, Barbara, Ian, Martha and Jack (at least before the whole "immortal" thing).
Ace can arguably lead the pack of the companions, being of a quick temper and always carrying with her a backpack full of homemade explosive charges that she's not afraid to use. The woman took out a Dalek with a powered-up baseball bat. And dispatched another with a rocket launcher.
The Doctor: You killed it! Ace: I aimed for the eyepiece.
From "The Next Doctor" Jackson Lake. An amnesiac Victorian gentleman who believes himself to be the Doctor and does very well, to the point of saving the real Doctor from Cybermen with a hastily improvised weapon at least twice.
A lot of the one or two episode based companions serve as this in comparison to the more seasoned TARDIS travelers.
In series six, there's Canton Everett Delaware III. If the name isn't enough, there's always the following: he tells off the President of the United States from the get-go, runs into the TARDIS without a second thought to keep on top of the Doctor, pulls off lying to the FBI for three straight months while presumably keeping Rory, Amy, and River out of their cross-hairs, shoots down a Silent, tricks said Silent into dooming his race, works future technology to send the message of the Silent to the Doctor and Apollo 11 and to top it all off, comes out to President Nixon in the Oval Office.
Rory Williams. Think he's just a nurse? You're wrong. It turns out that he, and not the Doctor, is the "good man" mentioned in the widely quoted saying, "Demons run when a good man goes to war."
While physically human, he "lived" for 2000 years as a plastic centurion, alone, to keep Amy safe. He infiltrated a Cyber-Ship demanding his wife, he withstood House and the Minotaur's Mind Rape, fought off the Headless Monks at Demon's Run, encountered more of the Silence over the course of three months than any of the other characters, and punched Hitler.
I'm here to kiss my wife or kick your ass. And I don't see my wife.
Semi-lampshaded in Power Rangers Operation Overdrive for the series' 15th anniversary Reunion Show. Four recent Rangers with civilian form-only extra powers were teamed up with Adam from season two. After a straight martial arts fight scene where the others used their powers, Adam commented on his unpowered martial arts skills, "It may be old school, but it gets the job done." Considering how the fandom has complained about civilian powers and the fact that the actor's face was always visible (to assure the audience that no stunt double was being used), probably overlaps with Fanservice.
Most every Ranger has some badassitude without their powers, showing them fight the Mooks of the season is supposed to help the transition to the Super Sentai costume footage.
Honourable mentions go to Takaoka Eiji from Go Go Sentai Boukenger. Before he becomes BoukenSilver, he is so badass that in his human form he can beat up two Ashu demons, whom the five other Boukengers together, in their Ranger form, were unable to defeat.
Choujuu Sentai Liveman: Arashi Busujima. His debut episode features him biting a Monster of the Week and turning it and its minions into his new gang. And the Liveman had a lot of trouble with this Monster of the Week, too. Also, in addition to not having special powers, the man is an idiot among geniuses. Sure, next episode he's turned into a genius, but he eventually is reverted back into his old self. At which point he make a suicide run into what was arguably the strongest Monster Of The Week in the season and succeeds.
Kung Lao, Siro, and Taja, the protagonists of Mortal Kombat: Conquest, are all ordinary human beings who regularly face off against supernatural opponents who can shoot ice, throw fire, teleport, devour souls, re-animate the dead, etc. Not only do they stand toe to toe against these enemies but they usually win. In one episode Shang Tsung, Kung Lao's primary rival, destroys a military encampment with his powers because he was just that pissed. Had he thought of doing that when he faced Kung Lao in Mortal Kombat, Earth Realm would have been doomed before the series even began.
When Quan Chi and Kung Lao first fought, Quan Chi tried to use his sorcery but was immediately stopped by Raiden in the name of "a fair fight."
Every character in the Merlin series (the one starring Sam Neill) who doesn't have magic is this. King Vortigern rips through a platoon of enemy soldiers, has dealings with the queen of The Fair Folk, and unflinchingly faces a powerful wizard in single combat. Frik manages to take down numerous opponents, survive a war, and work against Mab even after he lost all of his magic. Ambrosia regularly acts unimpressed by magical beings such as Mab, insults them to their faces, and declares that one point that, magic or no magic, if Mab harms Merlin in any way, "I'll have her guts for my bootlaces."
Logan of Dark Angel manages to come to the rescue of the genetically engineered protagonist more than once despite not only lacking superhuman abilities, he's also in a wheelchair.
Oliver Queen on Smallville isn't an alien, a cyborg, an Atlantian, or a metahuman. But between his training and resources, he holds his own rather well.
Same goes for Tess Mercer. She's not superpowered in any way. She's just verysmart and skilled.
The show's version of Hawkman, who's wings are instead a harness he wears to fly with, is easily one of the most badass supporting characters in the show. His only 'Power' is that he's cursed to be reborn every time he dies, grow up, fall in love with the same woman, and then die again. Yet, he utterly owned Green Arrow in their first meeting, and later saves Lois from falling to her death, after being stabbed and set on fire. It cost him his life but still.
Lois may be Overshadowed by Awesome a lot, but considering in her second episode she kicked the shit out of a trained soldier, and every time she gets into trouble she does so fighting and usually takes down a few mooks first, she definitely counts as a Badass Normal.
Elena Gilbert, protagonist of The Vampire Diaries, is one of an increasingly dwindling number of major characters who don't have superpowers, but she shows herself to be able to hold her own and gain control of situations where she is physically the underdog on numerous occasions.
Alaric Saltzman is a much better example, until he he gets turned into a vampire in Season 3
Bill and Jeremy also fall under this category.
On Merlin Arthur and the Knights of the Round Table.
Riderman from Kamen Rider V3. Unlike other Showa Riders who was a cyborg, Riderman was just an ordinary human with an artifical arm as enhancement. Doesn't stop him from being pretty awesome, though. His will and determination along with his own smarts and resourcefulness is more than enough to close the power gap.