Bleach: The team Urahara puts together to go and rescue Rukia in the Soul Society arc is one these: Ichigo may be The Hero, but he's only had ten days of training. Sado may be a powerhouse, but he's only possessed power for barely a month. Orihime has possessed power for barely any longer, but at least she can heal everyone, and Ishida is the archer whose got a score to settle with shinigami in general. None of them are experienced with either the power level they'll be facing or with real battle, but they go anyway. Because Ichigo's their friend, and Ichigo owes Rukia his family's life.
The Dollars gang in Durarara!! is surprisingly this, in spite of their sinister reputation.
And then there's Irresponsible Captain Tylor, whose crew is mostly composed of the kind of people you don't want near pencils for fear of what they might do to each other with them, much less a destroyer-class military space ship.
In the Soyokaze's case, the reason it qualifies is because the aging, broken-down destroyer has been assigned as the official dumping ground for all the lunatics, incompetents and misfits of the UPSF. In other words, every trouble-maker or disruptive element that accidentally manages to get into the military is invariably assigned here, so they'll be out of the way. The doctor is an alcoholic who's been drinking since he was three years old, the marines are all violent slobs, The Ace is arrogant and full of himself, as is the navigator, and the captain is, as far as the military higher-ups are concerned, either an absurdly lucky moron or possessed of genuine great insight but limited common sense. The only outright military and competent crewmembers are Lieutenants Yamamoto (who was assigned as the First Officer in the hopes he could somehow cover for Tylor) and Yuriko (who volunteered to join the Soyokaze in the hopes that she could somehow reform the crew).
The crew of White Base in the original Mobile Suit Gundam was comprised mostly of civilian refugees and a handful of junior officers who survived the attack on Side 7 in the first episode. They still manage to score a number of improbable victories against the elite forces of the Principality of Zeon, thanks to the Super Prototype principle and some of the cast developing into Newtypes. And more importantly to Federation command, they made really good decoys.
Both justified and subverted in 20th Century Boys. When Kenji starts up La Résistance, it's made up of guys he knew back in middle school, as they would be the only ones who were remotely familiar with who they're fighting against. After all, it's not very easy to recruit somebody off the street to fight against a cult based on your own twenty-year-old fanfiction. This ends up blowing up in his face for several reasons, the first of which would be that one of those Ragtag Misfits is the cult-leading Big Bad...
Eyeshield 21's Deimon Devil Bats. Other teams have full rosters, deep benches and long traditions. The Devil Bats only have 11 full-time team members (eight of whom were only just scraped up for this year, three by blackmail), and they all are weird in their own way.
The three helpers are also quirky. The two basketball players lent to the team and the miniature sumo wrestler. Though it feels like I'm forgetting someone...
Quite literally in Sayonara, Zetsubou-Sensei where the depressed-to-the-point-of-attempting-melodramatic-suicide-on-a-daily-basis teacher Itoshiki Nozomu's entire class, the very same people who are supposed to take care of the world of tomorrow, are a bunch of social misfits, mental cases, borderline psychopaths and WORSE... In the end, Nozomu himself comes across as a perfectly sane and socially functional man by comparison.
Team Dai-Gurren fits pretty well here. Kamina and Simon, two idiots who had never seen combat before were able to put up a fight against a powerful commander of the enemy forces (through use of stolen mechs they had no idea how to pilot), steal their mobile base, summarily defeat every major general they came across, along with the Big Bad. Their unit consisted roughly of around 30 or so (grew exponentially by the end) people who barely managed to survive on the surface, who were all fighting for a cause they believed in. The Rule of Cool was in full effect here, as well.
Division 2, the main cast of Patlabor, is made up mostly of police officers who were either Kicked Upstairs by their superiors or deemed too wimpy or too wild for the rest of the force.
The Yang Fleet from Legend of Galactic Heroes certainly gave this impression. It was first formed as the 13th Alliance Fleet, composed of new recruits and the remnant survivors of another fleet. Their first mission was occupying an invincible space fortress, which they succeeded at with ease. Ultimately, the Yang Fleet gathers up a colorful array of characters: an elite combat division of expatriate Imperials, a venerable, old Imperial officer in exile, a womanizing fighter jock, a bureaucratic family man, an ingenious orphan, rogue merchants from Phezzan - all of them led by a Bunny-Ears Lawyer who would much rather read history books than wage war, but who just happens to be one of the most brilliant tacticians in centuries. Dusty Attenborough and Poplan coin the phrase "foppery and whim" to describe the Yang Fleet's motivations in the face of incredible odds stacked against them.
One reason why Yang is such a frighteningly good commander is that he has a very keen eye for talent and will disregard virtually everything else, which explains why the Yang team is so lax when it comes to military protocol: Yang handpicked his subordinates because they know how to fight and how to lead, the rest is irrelevant.
In Fushigi Yuugi, the antagonist Seiryuu warriors are mostly battle-hardened, ruthless killing machines, a few of whom could conceivably take over Big Bad duties in their own right. The good guys? An Ordinary High-School Student, a peasant farmer, a Wholesome Crossdresser, a permanently smiling monk, a rageaholic bandit, a burned-out country doctor, and a young boy who initially refused the call because he was afraid. Oh, and the Emperor. Subverted in that five out of seven of them get killed, and they actually fail to prevent the god Seiryuu from being summoned. Good only triumphs at the end because of a Heel-Face Turn by the Seiryuu priestess.
The Muto Extermination Squad in Busou Renkin. Put together because the only leader who could keep them in line is a General Ripper, and they are on their important task because everyone else is dealing with a bigger threat. Especially notable because they are the antagonists.
The Varia in Katekyo Hitman Reborn!. The future arc shows Bel and Levi apparently wanting to kill Fran. This of course is followed by Bel sticking knives in Fran's back.
D.Gray-Man has this when you consider the people that the Innocence chooses for their users. The current Exorcists consist of: a former circus brat, a teenage girl, a swordsmen who is actually a test tube baby, a teenage historian with an eye-patch, his 80 year old mentor, a manic depressive woman who has lost 100+ jobs in her life, a bipolar man who spent his whole life in a castle, a blind guy to be fair he lost his eyesight on the job, a man raised by a brothel owner, an ex-con, a sentimental artist, a circus animal trainer, a perverted nine year old delinquent with the arguably most powerful/competent being the womanizing alcoholic who can't seem to go anywhere without piling up debts and carts around a dead woman. Not exactly the sort of Apostles of God you'd take comfort in having the task of saving humanity.
Sanzo's team and Kougaiji's team in Saiyuki both fit. "Ragtag team" is even used to describe the Sanzo-ikkou at one point.
Ginga Kikoutai Majestic Prince has Team Rabbits. They are the best in what they do, but it takes long for them to overcome their eccentricities and lack of teamwork. The fact that they are mankind's last hope against a vicious alien invasion does not help.
Justified by the Survey Corps in Attack on Titan because only the most brave, determined or insane join this branch that suffers the most and heaviest losses against the Titans. Nearly every elite officer has at least one quirk but because they are so good at their job, those quirks are mostly tolerated. Levi, the in-universe World's Strongest Man, is a Super OCD clean freak. Mike Zacharius has a habit of smelling people as a greeting, which can be used to detect Titans and he is the second strongest after Levi. Hange Zoe has an unhealthy interest in Titans, but there is a good reason she is a squad leader.
The Sasuke Retrieval Team of Naruto can be considered this, especially since 4 of the 5 members were considered dead-lasts in the Ninja Academy
The Mekakushi Dan in Kagerou Project is built of of 9 members: A moody invisible girl, a nature and animal loving empath, a chronic liar shapeshifter, a 1/4 gorgon fujoshi , a ditzy idol, a mischevious computer virus, a Hiki NEET, a 12 year old stalker and an Android with amnesia. The only thing keeping them together being their shared societal abandonment and goal to save their loved ones from the Kagerou Daze.