This is such a well thought out topic.
Sooooo many great examples.
The first one that comes to mind is
It’s Okay to not be Okay

Why is it that this prompt is getting harder and harder? Found families' notion is like a pillar in kdramas. Actually, all my favorite dramas feature an epic squad of different people coming together. Be it cohabitation, sharing the same goal, or being thrown together in a team.
If I go with the first one jumping to my mind, it will be Thirty but Seventeen. Watching those people come together as a family was a delight that can never be forgotten.

I just watched 30 but 17! I loved that entire drama. I love found families, and this was just such a great one!

I would add in the Bossam: Stealing Fate family and all the Doctor Romantic families.

Really?! Why are you doing this to me dbstaff!?
Obviously ANSWER ME 1988
But here we go again, because I love the found family trope too!

Honourable Mentions (in no particular order):

Age of Youth S1&S2
I’ll Go To You When The Weather Is Nice
Tale of the Nine Tailed 1938
Tale of the Nine Tailed
Vincenzo
Fight My Way
The Package
Our Beloved Summer
Let’s Eat S1-S3
Work Later, Drink Now S1&2
Prison Playbook
Welcome to Waikiki S1&2
Answer Me 1994
Hospital Playlist
Move To Heaven
It's Okay That's Love
Coffee Prince
Taxi Driver S1&2
So Not Worth It

There is a similar genre to this which is close a friendship group, but I’m not sure if it fits into the found family criteria.

Answer 1997
Dream High S1
Love Struck In The City
My First Time
Twenty Five, Twenty One
Weightlifting Fairy Kim Bok-joo

There are way more Kdramas I’m missing, but yeah at least this list helps me (and hopefully other beanies) when craving the found family genre.

CLOY and Thirty but Seventeen are high on my list, but the family of misfits, underworld denizens, and hurt kids that gathers around halmeoni in Just Between Lovers is my favorite. The kitchen-gangster crew in Wok of Love and the accounting team of outcasts in Chief Kim are also wonderful.

Huh, there's a common element there that I didn't notice until I wrote this.

This probably is my favorite sub-genre in Dramaland, and they do it so well! It's hard to pick one, I have watched and loved quite a few "found families".

Prison Playbook counts? The bonds among the main characters are so strong. Even some of them end the series being really a family (after a long-kept promise).

Honorable mentions:

Live up to your name.
The best hit.
Shopping King Louis.

This is impossible because there are just toooooooooooo many and it feels like a betrayal to choose one over the others.

The father/son relationship forged in the prison cell in Prison Playbook which resulted in the ‘son’ finding a place on his release and then coming back to the prison to collect the dad on his release date to take him to ‘their’ new home is the one I will go with.

Difficult choice! It's such a warm and nice trope in Kdrama world.

I really liked Be Melodramatic group. They all gain something to live together : no rent to pay, free babysitter, people watching over you grieving.

Bulgasal - not only is the "family" a motley crew of totally different characters, they also share all kinds of different connections to several past lives.

Mother- 8 year old Hye-na is being abused by her mother and boyfriend. Soo-Jun, Hye-na’s substitute teacher becomes aware of what is happening, takes matters into her hands and decides to become her mother. Hye-na blossoms under her care.

uh so many like Lookout, Flower boy ramen shop, Uncanny counter.
But the one that felt most like a unity and unseparable is Radiant Office.

I don’t know why, but upon reading the prompt, my immediate pick would be Sh**ting Stars. They have one of the best, if not the best, supportive colleagues and office cultures I’ve ever seen in Kdramas. Everyone, except a few, is so precious.

This is one of my favorite things that dramas do - the coming together as a loving and caring (family) unit. Like others have mentioned, the found family in Thirty But Seventeen is one of my favorites. I also loved this element in The Best Hit, Bossam: Steal the Fate, The Fiery Priest, Unlock My Boss, PegasusMarket... I could go on!!!

Currently, HeartBeat is doing this adorably, with the FL and ML looking out for each other, checking in on each other, and saying encouraging words to each other🥹

OOOOH found families, there are so MANY. It is funny because it is a kdrama trope, but it is one, no one ever complains about because it is so endearing. My choice will be Healer because they found it, lost it, and found it again. Plus Ahjumma!!!!

So many brilliant found families, but I'm going to go with Vincenzo. The way the Geumga Plaza residents didn't help an ex-mafia lawyer find himself, but all discovered they should have all been in the mafia with him. Glorious subversion, utterly bonkers, and I'm still smiling as I remember it.

Rebel: The Thief Who Stole the People. Found family is a core theme for this one! Bossam was nice as well---and even shares a family member with Rebel.

I'll pick Rainbow Transportation team in Taxi Driver.

I also love Mad Dog team.

Reply 97, because what could be better than your parents skipping your graduation to attend the adopted son’s graduation 🤭
But honestly @mistyisles first answer is most fitting, it’s hard to choose when this is one of the most favored tropes and dramaland gives us so many great ones.

This is one of my favorite things in kdramas so this is especially hard for me. I agree with @mistyisles that whichever one I'm watching is often my favorite of the moment (especially since I just finished The Uncanny Counter!), but in the end I'd be lying if I didn't answer Crash Landing on You.

Se-ri and her ducklings were so crucial to that story, and I think that @daebakgrits is right that Se-ri needed hers more than most other characters. The moment when Se-ri sees Jeong-hyeok in Seoul for the first time and thinks she's dreaming is breathless and great, yes, but the moment when the ducklings are dejected on a park bench because they almost got caught and Se-ri shows up and pulls them into a group hug (scolding Pyo Chi-su to join) had me in happy tears.

I'm going to say Run On because, by the time the drama ended, I loved the four characters and their dynamic so much that I was so sad I wasn't going to get to see them anymore. I could have watched them just doing nothing and hanging out.

So many great dramas with found family trope have already been mentioned. I would add
Mad Dog, Unlock My Boss, The Secret Romantic Guesthouse, and Chuno to the list.

Normally, I'd go right to the neighborhood crew in My Ajusshi, but I'm watching Liberation Notes right now and I'm mesmerized by the Liberation Club.

I love this trope! I think my favorite k-drama example is probably Mixed- Up Investigation Agency because it's so unique in terms of story structure as stays so firmly focused on the friendship dynamics between its gang of misfits. Runner-ups include Chief Kim, Hit the Top!, and Fermentation Family.

I’ll include Yoo Na’s Street .

Straight arrow Chang Man moves into a small apartment building owned by a gangster . A strange mix of gangsters, a pickpocket, a gold digger and more .

My pick goes to The Uncanny Counter. I think we argubly have 2 sets of found family in this one. First, So Mun (is that's name?) and his friends. Oh to have friends like that. Second, the counters.

Both relationships are so important and sweet and protective.

I hope even though the cast has new people, So Mun's friends are still apart of the new season.

This isn't really family and my recollections are very vague. Who watched Love Rain? The parents of the male and female leads were ex'es. Please correct me if I'm wrong, but I think they met again.

Kdramas have taught me that my father will land me with a mountain of debt, my siblings will try to kill me, and my mother will try to sabotage my happiness - that is if she survives her bout with cancer.

We are nothing without our found families !

So many good ones already mentioned, but imma go first with "Goblin"
and then "Fiery Priest"...that writer whose name escapes me at the mo' is all about the found family "Chief Kim" ( where I first learned to love watching junho eat) and "Vincenzo" (new crush, Taecyeon, both adorable and frightening in turn) ....boy that writer also seems to have a 2pm thing...

I love this concept, and honestly, one of the things that made me find my entertainment nirvana in K drama land. As I understand, found families are those who have your back outside of your immediate family. For me Our Blooming Youth will forever be my favorite. The story is about these individuals who learn to trust, love, forgive and grow with each other. Two of them are orphans, one of them is plagued by the trauma of being hunted by those he should be trusting, one of them trying to find his mojo against the dictates of the society that has already decalred him a failure....and when they all come together, they are magic. This drama is no where perfect and could have done with ether some good editing or some added episodes, but it's a story that spoke to me and the characters brought to life by the actors made the expereince beautiful.

(1) The neighbours in Reply 88 (both the adults and the teens). The neighbourhood essentially functioned as a family.
(2) Seri's North Korean family in CLOY

Quite a few of my “found family” favorites have already been mentioned—most notably, CLOY, Greasy Melo, and Prison Playbook. But the one that I found most compelling was in Melancholia. A maverick math teacher and two of her students form a close bond centered around her renowned mathematician father, whose brilliant mind has sadly been affected by Alzheimer’s. The three generations are brought together through a shared passion for mathematics. I loved this idea—and especially the fact two of the math geniuses were female, in a field dominated by men.

I believe there was a bit of a found family in Legend of the Blue Sea as well, if I remember correctly (?) Everyone gathered at Lee Min-ho's house.

This is a super old one but Bad Family, starring Kim Myung-min and Nam Sang-mi. A little girl loses her family in a car accident and the trauma left her with amnesia and being unable to talk so her uncle hires a bunch of people to play the various family members she lost. Of course, it's a group of oddballs who later come together as a family to protect the little girl.

I thought Do do sol sol la la sol had a good found family theme, but the writer then jumped the shark and screwed up the drama

CLOY. Hands down. I wept and ached for days when it ended because I had come to think of them as my own friends. I didn’t want to say goodbye. I can’t even rewatch because I don’t want to have to say goodbye again. But I think about them all the time. Like DaebakGrits said, Se-Ri’ found family is the strength of this drama.

On another note, a found family drama I rewatch annually is Itaewon Class. Saeroyi’s unwavering loyalty and belief in others simply warms my heart.

The first one that came to my mind was the iconic 30 but 17 family. But I think I have to pick The Fiery Priest family. That squad is so iconic that even behind the scenes they became a family.

Apparently most of the members of the cast were having a hard time and found comfort with that project. They became really close friends.
And I also love how crazy the ensamble was. Some families are formed with people that have something in common, or they know each other from a really long time, or they're just similar. But this group has some of the weirdest combinations in kdramaland: a fiery priest, cops, nuns, a corrupt politician, a gangsta... LOL I just love it.

Honorable mentions:
Hello Monster (two family members lost each other and go through hell to find each other again. This is the real found family here).
Terius Behind Me
Mr. Queen
Zombie Detective
Coffee Prince
Stars Falling From the Sky
Ho Gu's Love
Homemade Love Story
Be My Dream Family
Kill Me Heal Me (I kinda hate this show but the relationship between the multiple personalities and the siblings is adorable. They became one big family at some point, imo).

NAVILLERA (2021) (Netflix US). Start of with the wonderful pairing of great character actors Nam Moon-hee (mom Hae-nam) and Park In-hwan (dad Deok-chul).
The organic family consists now of older parents and three almost middle age children with spouses and one grand child.
It was wonderful to see how Chae-rok (Song Kang) was brought into the family fold. Deok-chul was even able to use his grandfather charm to soften up Ho-beom (Kim Kwon). One if my favorite dramas of 2021.

O/T. I much prefer actor Jung Hae-kyun’s performance in NAVILLERA as uptight oldest son Sung-san than his current totally over the top wacko performance as Chairman Lee Chan-joo in NUMBERS.

So many, but "STARS FALLING FROM THE SKY" aka "WISH UPON a STAR." was so darn charming with all the kids and their names.

This was so hard.... 🥲

Seriously, it was. But I have to go with the Doctor Romantic series. I know I love some others (While You Were Sleeping, Age of Youth, It's Okay That's Love, Move to Heaven), the Doctor Romantic series holds a special place in my heart.

It's okay to not be okay & The uncanny counter tied for first place.
and as an honorable mention, the way Gil Daran's family (love their chemistry) "adopts" Kang Kyung-joon/Seo Yoon-jae. Kyung-joon was an orphan and when they're trying to find a rock to represent him in the family *weep*.