Flinn

"Finn, even if we like each other, we're going to hurt each other."

- Flame Princess to Finn in "Hot to the Touch"

Flinn is the Het ship between Finn and Flame Princess from the Adventure Time fandom.

Canon
In "Vault of Bones," Finn teaches her properly how to dungeon crawl, but she finds his methods boring and tells him that she is not having fun. After Finn allows her to do things her own way, she proceeds to burn everything down with her elemental powers. When a Giant Goo Skull captures Finn, she rescues him by intimidating it. At first, Finn thought he was going to be burned alive because of how Flame Princess approached the Goo Skull, but after he is rescued, Flame Princess says she would never hurt her boyfriend, calling him her boyfriend for the first time.

At the end of the episode, Flame Princess thanks Finn for taking her to the dungeon and suggests that they go to another dungeon sometime.

In "Jake Suit," she invites Finn to her house to meet her less-evil family members. She totally does not even mind that he makes a fool of himself once she finds out it was for a bet with Jake. She believes that Finn will win and cheers him on, giving him the support he needs to survive the final stretch of Jake's gauntlet.

In "Frost & Fire," Finn and Flame Princess are interrupted by Ice King in the middle of a picnic. After Flame Princess beats up Ice King for insulting her, Finn starts to have weird dreams with the Cosmic Owl in it, but keeps waking up before he can finish the dream. Finn tries to get Flame Princess and Ice King fight by writing insulting messages about them and claiming the other sent it. This, however, leads to Finn attempting to apologize and to continue their relationship, but to no avail as Flame Princess leaves saying "...I need some time alone."

In "Too Old," it is revealed the couple had broken up, and so Finn makes one last attempt at Princess Bubblegum. It fails as usual, and he tells Jake he has to "patch things up with Flame Princess," so it is unknown if they get back together or not, as Jake ominously states Flame Princess could have already replaced Finn with a new boyfriend.

In "Earth & Water," Flame Princess has troubles controlling her emotions again after her breakup with Finn and ponders whether she wants to be with him again. She later befriends Cinnamon Bun when Princess Bubblegum leaves her unattended in the middle of a test and tells him that sometimes friends are "hard to understand" and that they are "just playing mind games." This may be due to Finn keeping all the ladies in confusion after reading the book Mind Games in "Reign of Gunters." Later in the episode, Flame Princess overthrows Flame King and became the new ruler. When Finn tries to apologize to her, she accepts his apology. He asks her if they are still going out. She says no but they can still be friends and that he can come by anytime as long as he is completely honest with her.

In "The Red Throne," after being secretly poisoned by her treacherous hand maiden and escaping Flame Lord, Flame Princess seeks help from Finn in order to save the Fire Kingdom. Despite apprising Finn specifically that they are not getting back together and that they are only meeting so that he can help her reclaim her kingdom, Finn tried to impress the princess with his improved swordsmanship and showed-off impulsively by charging into the Fire Kingdom, which annoyed Flame Princess. In the dungeon, Finn revealed that he thinks that Flame Princess still has feelings for him, to which she protests.

Finn continues to admit to others his love for her in "Love Games" and "The Pit." Despite her rejection, she does, however, still want to be friends even after his betrayal in "Frost & Fire." In "Breezy," it is revealed by Dr. Princess that Finn no longer talks to Flame Princess, indicating he no longer has feelings for her and instead harbors contempt (although in "The Mountain" when he sees her, she "kinda messed (Finn) up"), possibly since he thinks Flame Princess and Cinnamon Bun are dating.

It is likely that Finn continues to harbor genuine romantic affection for Flame Princess, as seen in The Comet. When the Purple Comet offers Finn a new mode of existence, she explains the crisis caused by the weight of this decision. She goes on to mention the various aspects of his human life he would be free from, the first is love, which is signified with a modern appearance of Flame Princess. This isn't to say that his feelings might not have changed at all, or that they are reciprocated by Flame Princess, but they likely still exist despite the conflict caused in himself by her rejection.

In "Bun Bun," Finn delivers Bun Bun to the Fire Kingdom. Flame Princess was excited to see him and invited him in for lunch. Finn apologized more sincerely this time, saying "I said I was sorry but I didn't fully understand what I had done wrong." Flame Princess was happy to hear that and said that he's grown up a lot. After this the two have fun as friends and are now seen casually together without any drama.

In "Happy Warrior," Finn was concerned that the supposedly dragon (in fact, the image of the Fire Princess) had eaten it, but after a while, succumbing to anger, he himself became a Fire Elemental. Lumpy Space Princess thinks that they have a chance to come together again.

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Trivia

 * In the original storyboard for "Hot to the Touch," Flame Princess was supposed to kiss Finn; however, the idea was dropped and reused for "Burning Low."
 * She refers to Finn as her boyfriend in "Vault of Bones" and Finn refers to Flame Princess as his girlfriend in "Puhoy."
 * In "Puhoy," Finn is briefly concerned that their relationship is over after Flame Princess did not get a joke he made, and ventures into a Pillow World where he lives through a lifetime with a pillow wife and pillow children. When he dies of old age in this world and "returns" to Ooo, he immediately gets a phone call from Flame Princess telling him she now gets the joke and promptly forgets about his "dream" of the Pillow World.
 * In the episode "The Pit," Finn mentions that he still loves her, even though she rejected him in "Earth & Water."