At a screening in San Francisco, Matt Ross revealed that over the course of filming, the group of children came to call Viggo Mortensen "Summer Dad".
Viggo Mortenson's crazy, awesome, red-patterned shirt that he's wearing in the funeral scene of Captain Fantastic is the same shirt he wore in the 1992 movie Indian Runner of which snippets were used in the Bruce Springsteen music video for the song "Highway Patrolman" (1982). In the video, Mortensen plays the bad luck, no good brother of the highway patrolman.
In the campfire-scene at the beginning you can see Rellian reading "The Brothers Karamazov" by Fyodor Dostoyevski. The novel is about three brothers, who -with increasing age- start to shun and rebel against the ways of their father. Just like Rellian himself does later on in the movie.
During several parts of the movie, Viggo Mortensen's character is seen drinking Mate, a traditional South American beverage. The actor lived in Argentina many years, so this is probably a nod to his own past there. Mate (pronounced Mat-ey) is also known in Middle-East Lebanon and Syria.
During a breakfast scene, Viggo Mortensen is seen drinking "mate", a typical argentinian infusion made of herbs. He lived in Argentina from age 2 to 12.
Vladimir Nabokov's "Lolita" was referenced in the movie by one of the daughters, the 1997 movie version of which Frank Langella played the character "Quilty". Frank Langella played the grandfather in this film.
This is the second movie starring Kathryn Hahn that features a family singing a rendition of "Sweet Child of Mine" by Guns N' Roses. The first was Step Brothers (2008).
In the final scene, Ben (Viggo Mortensen) has a flask with a sticker of San Lorenzo de Almagro, argentinian club wich he is big fan.
The song that plays on the supermarket loudspeaker is "My Heart Will Go On", the theme song from Titanic (1997).
In this film, actress Erin Moriarty shares a brief romance with the fictional brother of actress Annalise Basso. In the film The Kings of Summer, Erin Moriarty shares a relationship with Annalise's real life brother, actor Gabriel Basso.