Kyon from Haruhi Suzumiya. He does have one thing that sets him apart from everyone else: He's John Smith. This fact puts him in Badass Normal territory, because it allows him to control Haruhi, Yuki's boss, and almost anyone who knows of Bamboo Leaf Rhapsody. It also explains why Haruhi gravitated towards him: he's the reason she came to North High to begin with (she just doesn't know it yet).
Mahou Sensei Negima!'s Asakura; she's one of the first girls to discover Negi's magical abilities, yet never gained any sort of magic or kung fu abilities of her own until near the end of the series, and it was an emergency situation. Despite that, she was still able to be more or less self-sufficient when she was forced to fend for herself in the Magic World.
Jonny "Blazing Rumbling Trucker" Raidein is one in the Magic World arc, being nearly the only non-empowered person in the crew. He helps out Ala Alba purely because he's friends with some of the girls, and takes the whole "saving the world" thing in stride.
Bobobo-bo Bo-bobo: Despite the extreme, glass-shattering reactions to her companions' antics most of the time, Beauty has no issue with staying with Bo-bobo and his downright insane group of rebel fighters; in fact, she embraces her life with them with joy. The end of the sequel manga has her mention that she rejoined Bo-bobo for another adventure, despite the danger, because she had grown to love his insane, comedic lifestyle.
Ataru Moroboshi of Urusei Yatsura. Though, this might be subverted as time goes on, as he seems to gain an inhuman level of super-speed (whether running away from trouble or running after a woman) and a comic invulnerability that would make Daffy Duck proud.
Nabiki Tendo of Ranma ˝. For the most part she's unfazed by the gender-bending, shape-shifting, and martial arts insanity surrounding her, getting involved only to the degree that she can make money from it. Her sister Kasumi is an even better example. She's never even phased fazed at the insanity surrounding her family, to the point where she constantly refers to Ranma's various mortal enemies as his "friends."
Haruhi Fujioka, from Ouran High School Host Club, who starts out utterly weirded out by the bishonen antics of the aforementioned club, but gradually learns to cope, even enjoy herself a bit.
Naru Osaka, from Sailor Moon, in both senshi and youma knowledge. It rendered her the nickname of Youma Bait; to a lesser extent, Umino fits this trope.
Played with in Fullmetal Alchemist when Ed finds himself surrounded by Homunculi and chimeras and humorously realizes that he, as the only normal human, is the minority, even though it's hard to consider cybernetic limbs and alchemy as normal.
Oz from Pandora Hearts takes pride in his ability to accept the oddities he encounters and adapt to them. This manga is one big Alice AllusionMind Screw, so having that kind of adaptability deserves some applause. Subverted when his true origins are revealed.
In Batman, AlfredPennyworth is an example of this trope. His duties as butler at Wayne Manor include cooking, cleaning, laundry, tending the -often serious- injuries of the Bat-Family, maintaining crimefighting equipment and sometimes using firearms to defend the Manor and Cave. He does this all without losing his composure, wrecking his suit or missing tea.
...and his Marvel Universe equivalent, Rick Jones.
Jeremy Feeple from Ninja High School is no doubt this, especially considering the weirdness he attracts to himself.
From his own point of view, Howard the Duck is one of these for the Marvel Universe. True, he is an anthropomorphic duck, but he comes from a world where that is normal so living among humans does seem like being surrounded by strange aliens to him. What is more, his attempts at leading a normal, blue collar life are constantly being interrupted by encounters with people who are even considered strange by Earth standards, such as the Man-Thing and the She-Hulk.
Agent John Meyers in the first Hellboy film, an FBI agent who gets transferred to the "nonexistent" Bureau of Paranormal Research and Defense. He's chosen by Professor Bruttenholm specifically to be Hellboy's conscience and moral helper. Too bad he's Put on a Bus in the sequel.
Janine Melnitz from Ghostbusters. She is surrounded by men who hunt ghosts, half of whom are mad scientists. She even has a ghost in a cage close to where she works that the busters keep as a "pet". Not once has she ever batted an eye at any of this.
Winston Zeddmore was this in the first film, casually commenting "As long as there's a steady paycheck in it, I'll believe anything you say."
The Love Interest from Red. Sarah is kidnapped, drugged, shot at, almost blown up, and kidnapped again. She is surrounded by old and young spies, assassins and government agents who are fighting over her life all the time. What does she say when she is confronted with the death sentence or life in prison if she is caught? "Awesome." And when she finally is?That her boyfriend will kick the interrogators ass. The movie is full of lines like this from her.
Francesco Dellamorte, Cemetery Man - for some reason, unbeknownst to both the townsfolk and the audience, all corpses in the Buffalora cemetery reanimate seven days after burial. Despite having no clue as to why this happens, Francesco expresses neither surprise nor concern, and instead dispatches zombies nightly without batting an eyelid. He refuses to inform the authorities of the situation because he does not want the cemetery closed, which would put him out of a job.
"This is my business. They pay me for it." "There comes a moment in life when you realize you know more dead people than living." "Disposing of dead people is a public service, whereas you're in all sorts of trouble if you kill someone while they're still alive." "Hold on a minute Franco - *shoots zombie* - You were saying, Franco."
There's also the janitor played by Harry Dean Stanton who helps out Bruce Banner. Coulson, at least, is a full-fledged secret agent - this guy sees a giant naked green man crash through a warehouse ceiling, and decides to find him some pants once he goes back to normal.
Not alien or fantastical, but Slevin Kedevra in Lucky Number Slevin shortly after the intro gets pulled into the world of mob bosses and hit men (by being mistaken for a dude that owes them a lot of money), but acts completely unfazed by most of his ordeals. Subverted though, in that he intended for all this to happen, so he could enact his revenge upon the mob bosses who killed his father.
Not alien or fantastical either, but Keiko Nishi in The Big Hit goes from being a normal everyday heiress to enthusiastic participation in her new boyfriend's high-end assassination business and takes being kidnapped, bounced around in a life-or-death joyride, and other weirdness pretty much in stride.
Mills of The Living Wake is a bit of a Cloud Cuckoo Lander, but compared to his surroundings, including his Upper-Class Twit of a best friend who has until exactly 7:30 that night to live, he's quite ordinary. He doesn't appear to mind, though. Organizing and Attending Your Own Funeral? Well, if that's what K. Roth wants! Random musical numbers? Just another day! Smuggling horrible and disturbing children's books into a library? Sure, why not! Distracting a ninety-year-old man so K. Roth can go on a date with his wife? On it!
Agent Kay in the first  movie was more than willing to accept that aliens exsist, are in fact all around us, enough so to join the Men In Black. What he doesn't get is Agent Jay's behavior and quick use to the memory eraser pen.
In the original radio series, he progresses to Badass Normal over the course of the Brontital storyline, even holding his own in a firefight with the Footwarriors (granted those guys couldn't run very fast), and by the end he steals the Heart of Gold with his girlfriend and swears a blood oath against Zaphod to avenge the destruction of Earth.
He also actually takes some time to actually come into the "coping and rationality" part of the trope. In the movie, he actually comes close to cracking under the weird a few times.
In the books, he actually decides that he will go mad. And then he does. Although that was due more to loneliness than the weird, the weird still played a major part.
"The weird" actually ramps up somewhat at that point: Shortly after deciding to go mad, Arthur finds himself chasing a sofa across the fields of prehistoric Earth, which pleases him greatly at how well the madness thing seems to be going for him.
Jason Wood in Digital Knight. By the end of the book, he has fought, befriended, and had interaction with vampires, werewolves, demons, and basilisks. He even married a witch.
Jim in Skunk Works
Kit's older sister Carmela in the Young Wizards series, though she's a secondary character who doesn't get much "screen time" until the later books.
And then she jumps over to Badass Normal with a mail-order death ray.
Bilbo Baggins in The Hobbit. He lives in a world well known to contain wizards, dragons, dwarfs and the like, but he, like most hobbits, has spent most of his life comfortably away from all that and is certainly dragged into it. note None other than the former trope namer's actor (Martin Freeman, not Simon Jones) has played him in Peter Jackson's film adaptation. Given that Freeman also played Unfazed Everyman Dr. Watson in the modern-day transposition of Sherlock Holmes, Sherlock, he really is in danger of being typecast.
The first of John Brunner's The Traveller in Black stories has a 20th-century Londoner called Bernard Brown find himself, for some reason which is never fully explained, trapped in the stories' setting of a far-distant past where magic still works. He makes his way to the nearest city where he gets treated as a god. He manages to defeat the real god of a rival city by using the kind of calm and logical approach which the people of his adopted city prize most highly.
It's fairly strongly implied, though never stated explicitly, that the Traveller brings Bernard to the city so that he will do precisely that, thereby sating a sudden mad desire on behalf of the people of the city to have a god — any god — and returning them to their normal level-headed ways. This unexpected change in the city's collective behavior can be interpreted as a move made by the forces of Chaos, but that requires reading between the lines.
It's the explicit granting of a wish to teach a lesson. It's the Traveler's way of dealing with bad situations by being a literal genie to those who ask him for favors.
Waldo Butters, in The Dresden Files. Well, he's a muggle who accepts the supernatural easily and hangs around with a powerful wizard who's always getting him into danger... his freakish love for polka might discount him from being considered "normal", exactly.
Lyra Volfrieds from Black Dogs. Starts out as an unathletic bookworm, wide-eyed and surprised at everything from dog soldiers to lesbians.
Rincewind the "Wizzard" of Discworld fame. He is such a bad wizard that if he died, the per capita magical capacity of the Disc would go up. Despite his ineptitude, he has been to hell, heaven, Earth, the dawn of ages, the court of the fabulously rich and good-natured serif Creosote of Klatch, the court of the fabulously rich and Obliviously Evil emperor of Agatea, the abode of an Eldritch Abomination, the Dungeon Dimensions (from which the latter originally came), Death's Domain, and outer space, and not necessarily in that order. Although he is a wizard, and can still, for example, see Octarine (he does not know the time of his death, but that is because his timeline is so screwed up that even Death doesn't know, rather than any incompetence or tendence towards muggledom).
Faile in The Wheel of Time is introduced as one of these. She gets more nuanced as the series progresses, but she is introduced as a normal person who just gets caught up with a cast of strange characters, much to her initial annoyance.
Alice in Alice in Wonderland and Through the Looking Glass. Though her adventures regularly give her fairly concrete evidence that she's gone out of her skull, she takes it pretty well, given that her standard reaction is generally along the lines of "Golly, that's unusual", followed by a brief philosophical discussion with said unusual thing about whether or not she's objectively justified in considering it unusual.
The protagonist of The Subject Steve pulls this off, despite a lack of supernatural elements in the book.
David Wong and John Cheese from John Dies at the End are Unfazed Everymen who come into contact with all kinds of weird shit because of the soy sauce and eventually become unaffected and Genre Savvy.
George Dorn from the Illuminatus!-trilogy definately qualifies: he gets recruited to the League of Dynamic Discord implicitly due to this quality, and promptly sent to deal with the International Crime Syndicate for a job vital to the safety of the human race after being member only for a day or two, and only getting a brief explanation for the goals and history of the organization. Though it's not directly stated, it's implied that Hagbard Celine does this because he wants a person who's mostly free even from the Discordian preconceptions to be his representative. In fact most people in the Discordian organizations are like this when they first join.
Kamele Waitley from the later Liaden Universe books by Sharon Lee & Steve Miller, who over the course of knowing Professor Jen Sar Kiladi goes from being a naive, cloistered ivory-tower academic (Fledgling, Saltation) to setting out to "rescue" Kiladi from Clan Korval but ending up rescuing herself from the machinations of the Department of the Interior instead (Ghost Ship, Dragon Ship).
Coraline is an Unfazed Everygirl; admittedly, she does get scared (as who wouldn't) by things like a demonic "Other Mother" attempting to replace her eyes with buttons, but she always maintains a calm, even tone.
Henry Bacon from The Last Dragon Chronicles - he never really knows what's going on, but he provides sturdiness in all the chaos surrounding him.
Ed from The Tome of Bill. Though he does react to the big things, he takes most everything else - including discovering that Bill is a vampire - in stride.
Comes up quite often in Animorphs, usually people whose cultures have myths of animals turning into humans (or vice versa). They don't seem all that concerned that a bunch of monkeys or seals or whatever turned into teenage humans.
Rachel Elizabeth Dare from Percy Jackson is a human gifted with Sight (the ability to see through Mist which hides the Mystical). she eventually uses this sight to become the Oracle.
Live Action TV
Many of The Doctor's companions in Doctor Who. Ten, who loves pop culture, notably considers Arthur Dent a "nice chap".
It has been lampshaded (in a rather dark way) that being one of The Doctor's companions turns you into a Badass Normal.
One of the stand-out moments being in "Delta and the Bannermen". After an initial shriek of fright, Mel is seen nonchalantly sharing her room with Delta and her newly-hatched green alien baby.
Rory Williams really wants to be this trope. By the time he reaches it, however, he's turned into a Empowered Badass Normal in the process.
His dad however, is. Brian Williams gets picked up by the Doctor by accident, and of course is a little miffed at first. After that, he goes along with the Doctor not questioning anything.
Gwen Cooper in series 1 of Torchwood, and her husband Rhys Williams from series 2. In season 4, Gwen scares a CIA agent by being rather unfazed after a battle.
Marilyn in The Munsters. Of course, Marilyn has the distinct advantage of being raised by, well, monsters. She believes that monsters are completely normal, and that she's the freak of the family.
John Watson from Sherlock. Yeah, every now and again his military and medical expertise comes in handy, but he's usually just a Badass Normal doctor; yet, he willingly follows along on bizarre and dangerous cases, and is probably the only person in the world who not only tolerates the Nightmare FetishistInsufferable Genius that is Sherlock Holmes, but also considers him his best friend.
John Crichton in Farscape, initially, though after a few years of being the universe's punching bag, he becomes Badass Normal on the grounds that whatever doesn't kill you makes you stronger. He may qualify as an evolved Unfazed Everyman after the Ancients unlock the wormhole knowledge in his brain. Though not Genre Savviness, wormhole technology is apparently the most powerful tool available to the the major characters. By the end of the miniseries, he creates an exponentially expanding wormhole capable of destroying the entire Universe.
Oz also had the potential for this before going all wolfie. His reaction to discovering the existence of the supernatural (though he did react like that to everything...):
Oz: Yeah. Hey, did everybody see that guy just turn to dust?
Willow: Ohhh, well ... sort of.
Xander: Yep. Vampires are real. A lot of them live in Sunnydale. Willow will fill you in.
Willow: I know it's hard to accept at first.
Oz: Actually, it explains a lot.
He has more of a reaction to Willow being with Xander than he does to:
Helping find Buffy after she was turned into a rat, who then turns out to be naked when she's changed back.
A student who's turned into Mr. Hyde wanting to kill him.
Willow appears to have become a vampire.
Buffy can read his thoughts.
Stiles in Teen Wolf. Despite being one of the only humans in the central group, he's remarkably unfazed by all of the weird stuff around him and at times is even less surprised by supernatural things than the werewolves themselves.
His only complaint about his best friend's werewolf transformations is that they're 'very stressful' on him.
Scott: I swear, I don’t have the urge to maim and kill you.
Stiles: You know, you say that now. But then the full moon goes up and out come the fangs and the claws and there’s a lot of howling and screaming and running everywhere, okay? It’s very stressful on me! So yes, I’m still locking you up.
Ando Masahashi from Heroes was one for a while, but eventually he got powers of his own.
HRG's wife and Claire's adoptive mother, Sandra Bennett, seems to have adapted to her daughter having superpowers and her husband being an all-purpose spook rather well, all things considered.
Frank Lapidus on LOST, a recurring character introduced in season four (and later made a regular for season six). He doesn't have any connection to the story or mythology, and his only personal problem (being a drunk) was dealt with before he arrived. He's possibly the only normal, well-adjusted person to ever set foot on the Island...and also accepts all the weird crap that happens on it without question.
Wendy from The Middleman was hired for her cynical, snarky attitude and matter-of-fact reactions to things like the eyeball monster.
Sheriff Jody Mills from Supernatural who fights the supernatural even though she is not trained as a hunter.
Andrew of Dino Attack RPG is just your average civilian who hangs out with Laxus and Pterisa while fighting an apocalyptic war against Mutant Dinosaurs, all without so much as raising an eyebrow.
Johnny Cage is this for the first story mode chapter of Mortal Kombat 9, by way of being the most relatively normal kombatant.
Derek Badger, unflappable Yorkshireman and protagonist of Ben Croshaw's Poacher. Not even a tumble down the proverbial rabbit hole into a world of spirits and monsters can shake him.
Dudley from Street Fighter plays the role in Street Fighter X Tekken. Elena shows up on his doorstep and tells him that a tree in Dudley's garden has told her about a great danger, so they must go to the South Pole and stop it. Dudley's reaction? It's all right, he'll help her in her quest, as long as he's back home in time for his next boxing match.
Dad Egbert from Homestuck. As the only naturally born human in the comic he was never destined to play a part in Sburb at all. His involvement in the game and very existence seems to be a complete accident.
Moloch VonZinzer of Girl Genius is an ordinary mechanic and soldier fallen on hard times, who against his best wishes, keeps getting caught up with Mad Scientists of all flavors, to quote "I've been around way too many sparks!" Even when he tries to stay out of the way ("I'm nobody's happy helper, got it?!"), he's clearly minion material, as he immediately accepts the job of carrying the Sparky heroine's tools and (unwillingly) following her through death traps.
Although sparks have a natural, almost supernatural charisma that makes anyone not Badass Normal into a willing (or in Moloch's case, unwilling) helper. Especially with Agatha, who is one of the most powerful sparks in the series.
Airman Higgs at first seemed to be one of these - a common soldier being hauled along after a group of very important people and managing to maintain some degree of detachment. He's actually a subversion, though, as bit by bit he has revealed various superhuman abilities. He is incredibly durable, superhumanly strong and quick, personally acquainted with Castle Heterodyne, and may either be connected to the Jagers or actually be one.
Sarah:Thanks, but I'm really not that cool. I just have some experience with this sort of thing...
Digger, of the Ursula Vernon webcomic of the same name, is an anthropomorphic wombat who manages to be an Unfazed Everyman thanks to the sheer weirdness of the rest of the characters. In fact, this is practically Digger's racial superpower: wombats are so sensible and tied to realism rather than faith or magic that they gain a mild immunity from divine/magical/prophetic dealings.
Sam: I was once like you...I had no idea there was an entire world of weird stuff out there. But then, bit by bit, my life got caught up in it. Soon, just dealing with it was practically a full time job! Devahi: What did you do? Sam: I made dealing with it my full time job. Fuzzy: Here's our card!
While having an identity crisis, Kronos from Wayward Sons befriends a family of feudal Chinese peasants, and brings them back to the spaceship to meet his crew. The father of the family, Huang, seems considerably less worried about the monstrous-looking aliens than he was about local (human) warlords, and his bio notes that he's good at finding things he can relate to about them and makes friends easily.
Bee of Bee and Puppycat has a tendency to under-react to events that would freak out lots of people. The most she gets fazed is frustrated groaning when a letter magically appears out of nowhere. Upon reading it, and seeing it offers her new pet cat and/or dog temp work, she offers to get him a pen to sign the letter.
Although the fact that he was conceived as his own grandfather giving him no delta brainwaves, and deliberately sent into the future by Nibbler specifically because of this may disqualify him.
The humans in all series of Transformers. They exist to not be awesome next to all of the cool robots. In fact, the most annoying humans have been some of the ones designated to be more equal to the Transformers themselves. Except for Sari, but she's unusual.
However, the fleshies of Transformers Prime seem to hold up quite well next to their co-stars, as each has perpetrated his/her fair share of badassery.
Ben and Gwen Tennyson in the original Ben 10, pilot episode. In fact they only freak out for a couple minutes, as Ben had turned into a living candle and had accidentally started a forest fire. After Grandpa Max explained that Ben was an alien they calmed down quickly.
Julie Yamamoto in Ben 10: Alien Force, while she only finds out in the last 5-10 minutes of the episode she found out in. She had caught Ben in his Jet-ray mode and when questioned she said she thought it was "cool". And then she gets a pet that doubles as a Cool Ship.
In The Fairly Oddparents, the Poorly Disguised Pilot for Crash Nebula reveals the eponymous hero was like this once: the only human in an intergalactic school. Once he gets the powerful space suit he begins to get less so.
Mark from Ugly Americans could probably out-Unfaze Everyman Arthur Dent.