In "My Two Favorite People", Finn, Jake, and Lady Rainicorn encounter a wizard who is handing out power rings. Finn also wears 10 rings his hands with each one displaying a different power, just like the Iron Man villain The Mandarin.
The Lich King is a horned, ridiculously scary, skull-faced monarch who can raise skeletons and whose power lies in a pit of green ooze. Compare to The Horned King, a horned, ridiculously scary, skull-faced monarch who can raise skeletons and whose power lies in a pit of green ooze.
The game Dropball, played by Ghost Man in "Blood Under The Skin," is a Shout-Out to Bullet Ball, the high calibre tablegame designed for the 21st-century lifestyle that went memetic after creator Marc Griffin's memetically depressing/hilarious appearance on American Inventor. Ghost Man's voice sounds quite a bit like Marc Griffin's and several of his lines are quotes from the American Inventor episode.
The episode "No One Can Hear You", references Alien when the candy people are shown to be stuck to the walls of the sewer with a goo, like the Xenomorphs do the their victims. Even the title is borrowed from the tagline of the movie: "In space, no one can hear you scream." Bonus Points for Finn and Jake screaming in the title card.
On the opposite end of the spectrum, Finn's casts breaking is really similar to Forrest Gump's leg braces.
The music box Finn uses to accompany the lullaby is strongly similar to the Orb of Isis in The Simpsons episode "Lost Our Lisa."
The Bucket Knight from "Dungeon" is a shout-out to the Alternian Imperial Drones from Homestuck. The artists are confirmed to be fans: a picture of their whiteboard shows doodles of the main characters.
Well, seeing as the comic itself was written by Ryan North...
The episode "Wizard Battle", the Grand Wizard states that wizards will fight using the eight schools of magic. This is likely a reference to Dungeons & Dragons' eight schools of magic.
In the same episode, the Wizard Abracadaniel shouts "Cantrip!" upon casting. In D&D, a cantrip is a minor spell typically only useful as a distraction in combat (incidentally, exactly the kind of magicAbracadaniel uses.)
In "Dad's Dungeon", Finn and Jake come up to a fork in the passage. One path has magical flowers. The other has a giant monster that says "None shall pass!"
In "Five Short Graybles" has Princess Bubblegum making a familiar toothy smile while enthusiastically saying "CHEEEESE".
"Princess Monster Wife" alludes to Perseus and Medusa when Finn and Jake use mirrors in order not to gaze upon the titular monster's hideousness directly, though they are only in danger of fainting instead of being turned into stone.
While trying to reassure Monster Princess that she's normal, Ice King refers to her as his "beautiful princess bride."
In "Wizard", Finn proclaims that he wants ALL THE STAR BADGES and strikes a pose similar to the "All the Things!" meme made famous by Hyperbole And A Half.
Earl of Lemongrab is an allusion to Mary Shelley's novel "Frankenstein." Like Frankenstein's Monster, he is a science experiment gone wrong who is rejected by his creator and forced to live in isolation. Lemongrab is also an allusion to Lovecraft's "Herbert West: Re-Animator," in which a mad scientist brings to life corpses, which become screaming, unintelligent psychos after he injects them with chemicals.
His name is a pun on Lemon Drop candy.
In "Sons of Mars", Abraham Lincoln gives Jake (as Magic Man) two choices of execution. The first is total annihilation, the second is convert his body to living stardust. During the second description, his heads floats amidst color bars that looks like "Jupiter and Beyond the Infinite" in 2001: A Space Odyssey.
In "Lady & Peebles", Ricardio closely resembles Majora's Incarnation. That part of the episode's storyboarder, Rebecca Sugar, is known to be a Zelda fan.
Also from that episode, PB gives Ice King a new heart partially made of maracas, which he then shakes while speaking vaguely Spanish phrases. This is possibly a reference to Charo, a Spanish-American actress probably best known for appearing on The Love Boat and "shaking her maracas".
It also (rather blatantly) references Hamlet with the acting troupe and Jake coming up with the plan from the play by himself to find those who plan to kill the Fire King. A portion of Hamlet's "To be or not to be" Soliquly is heard earlier as well, and it turns out that the Flame King played the Cladius, with the two plotting against him being would-be Hamlets, the Flame King's nephews whose father the Flame King killed to take the throne. This is common knowledge, and no one minds because everyone in the Flame Kingdom is evil.
The leader of the Destiny Gang in "Finn the Human" looks an awful lot like Kamina.
Speaking of anime, let's play a drinking game. Drink for every Fullmetal Alchemist reference. So, let's see. Two blond brothers, the older one has a mechanical arm; the mechanical arm can become a sword; the family household bursts into flames; untold power coming from red stones at the cost of thousands of lives; the forceful creation of alternate universes in which one is based in a rural farmtown...
Or on a creepier note, his utter ruination of their lives and the off-kilter, creepy "Playing God" vibe of the episode seem a bit reminiscent of a certain animation for Mark Twain's "The Mysterious Stranger".
Princess Bubblegum experimenting on some blob "people" by snipping off their limbs in "The Lich" could also be a reference to "Mysterious Stranger".
Marceline riding the Rock Giant's finger to the music shop nearly resembles the opening sequence of Superjail!
In Adventure Time: Hey Ice King! Why’d you steal our garbage?!! if you try to enter the first dungeon before finding Finn's sword, Jake tells Finn that "It's dangerous to go alone." a famous line from the original The Legend of Zelda.
One of the creatures, Lemon John, has a crisis: if he attacks the Candy Kingdom the Lemon people will live but the Candy people will be destroyed; if he doesn't the Candy people will live but his "family" will die. Lemon John states this in a way that sounds like Jean Valjean's "If I speak I am condemned/If I stay silent I am damned". Lemon John takes a third option: he saves everyone by reverting back a ton of lemon candy.
In "The Great Bird Man" Xergiok is reminiscent of Oedipus - he's a former king who's blind, lame and a recluse after committing great wrongs and his beard is "adorned with eyeballs", only this time they're whole.
Xergiok's eyes may also be a reference to Homestuck. They match (Homestuck spoilers) Sollux's after Eridan blinds him, and he explains their loss as, "I miffed off a wizard and he took my eyes."
"Simon and Marcy" has Simon singing the opening theme song of Cheers. "Filmed before a live studio audience. Norm!"
It also possibly pays homage to 'The Road' as well as some zombie movies such as '28DaysLater'
Probably unintentional, but the episode also seemed to have a few vague similarities to trailers for The Last of Us.
Choose Goose? More than likely a subtle shout out to the Goo-Goose that chews goo in Dr. Seuss' children's tale Fox in Socks.
In "Puhoy", Finn comments that Jake's pillow fort is "Smaller on the outside", which are the words used in the Doctor Who Christmas Special "The Snowmen" by current companion Clara Oswin Oswald.
When Rosalinen realizes Finn must return home, she is accepting of his decision and only asks that he remember her - the same request Marin makes of Link when she realizes he must leave by waking up, which will destroy her and the rest of Koholint Island in The Legend of Zelda: Link's Awakening.
As well as classic 70s kung fu movie references, "The Chamber of Frozen Blades" was basically one big Naruto reference. Perhaps most blatantly, the promotional art◊ shows Rebecca Sugar (one of the writers/storyboarders) wearing a Konoha hitae-ate.
In "Another Five More Short Graybles", Lemongrab shouts "ONLY ONE!" as he eats his brother.
"James Baxter the Horse" is a homage to a Disney and Dreamworks animator of the same name. Finn looks up to the fictional James Baxter like how Pendleton Ward looks up to the real James Baxter - who animated and voiced his namesake.
When Finn, Jake, and PB enter Wizard City in Wizards Only, Fools, it looks like Diagon Alley.
Also in "Wizards Only", the underwear thrown on Finn in prison might be a shout-out to the less child-friendly assault on Clarice in The Silence of the Lambs.
That episode is also as close as you can get to a Superjail! episode.
In "Be More" when BMO goes to get fixed, he notes their personalities and memories are erased and he doesn't want them to disappear like tears in the ''oven''.
Not really a shout-out, but "Björk" is used as an exclamation thrice in "Jake Suit".
One of the stickers Cinnamon Bun uses for his note in "Earth and Water" looks like one of the smaller creatures from My Neighbor Totoro.
In the same episode, after Flame Princess has deposed her father, she's wearing an outfits and hairdo that looks a lot like Azula. From Fire Princess to Fire Princess.
In "The Vault" The Regression sim game looks a lot like Super Hexagon
In "Memories of Boom Boom Mountain", the Head Marauder calls Finn a "chicken" as he walks out of the village to see who is crying. Finn slowly turns around with a Death Glare on his face and asks: "What did you call me?" This scene is VERY reminiscent of Marty McFly's reaction to being called the same thing in the Back to the Future trilogy. The fact that the Head Marauder is voiced by Thomas F. Wilson (aka. Biff Tannen) certainly helps.
In "Too Old", the original Lemongrab's appearance as a bloated Caligula figure that moves around by floating is very reminiscent of Baron Harkonnen.
In "Betty", During the fight against Bella Noche, a character named James (as in Jesse James) Longman, wearing a pink shirt, uses heavily slang-ridden speech, and shows a formula he made in his lab, which is not ideal. In other words, he’s a Shout-Out to Jesse Pinkman.
In "The Red Throne", the fight scene between Flame King and Don John the Flame Lord is a near-exact replication of the memetically infamous fight scene from They Live! (To top it off, both characters are even voiced by Keith David and "Rowdy" Roddy Piper, respectively)
One of Johnny's records in "Bad Timing" (specifically the one LSP uses for a gag) is the Frank Zappa album "Apostrophe."◊
Princess Bubblegum wears Dr. Sattler's outfit in James II.
The Fleshing out of the Lich due to the Crystal Guards blood is somewhat reminiscent to Eri Masami's demise in Serial Experiments Lain.
Also the Headless Horseman in Sleepy Hollow (which is probably a shout-out too given how much it's influenced by Hammer Horror movies).
The origin of the Gumball Guard in "Rattleballs" is a shout out to the Manhunters from the Green Lantern mythos. They even have the same color scheme as the Manhunters.
"The Tower" looks like it's made out of Minecraft blocks.
In "Furniture and Meat," Wildberry Princess attempts to execute Finn and Jake for their abuse of the power of wealth by ironically drenching them in their molten gold, likely imitating the Karmic Death of Viserys Targaryen in the A Song of Ice and Fire / Game of Thrones franchise
The "freaky clothes" Lumpy Space Prince dons in "The Prince Who Wanted Everything" are portions of the outfit of Char Aznable of Mobile Suit Gundam. They match well with his many Animesque facial expressions.
The 5th season (2012) episode "Jake the Dog" contains an author Shout out to the Head of Story, Kent Osborne. He was in a 2010 film called Uncle Kent, and the dialogue that Prismo and Jake share in the jacuzzi is verbatim that which Kent and Joe have in the swimming pool (the lines about co-ordinating breakfast and views on relationships)
During "We Fixed A Truck", BMO briefly shows the famous "Flying Toasters" screensaver from After Dark on its screen.
The season 6 episode,"Something Big", had a shout out to Attack on Titan, of all things, with giant humanoids, a flying army attacking said giants (the majority being killed in the process), cannons firing from the Candy Kingdom's walls, with the giants regenerating their limbs and heads when destroyed. There's even a colossal giant leading them, with a weak spot near its head, to top it all off.
"Breezy": Another Sailor Moon-esque transformation sequence.
The Rat King bears more than a passing resemblance to Emperor Zuul of all things. (They even fight in an area that resembles the background of their final fight in Super Robot Wars Z.)
"Something Big": Darren summons help during the assault on the Candy Kingdom with the incantation "I call forth the Legion of Cadmus, The Dragon's Teeth." This is a shout out to Greek Mythology, where Prince Cadmus threw dragons' teeth on the ground and had skeletal soldiers appear to fight.
"Princess Day" is a big honking reference to Thelma & Louise, complete with the iconic ending.