Gunter

Gunter is the name of the penguin that most commonly accompanies the Ice King, although in "Princess Monster Wife," it is shown that multiple penguins have that name but the Ice King pronounces the other Gunters' names differently. It does seems like trhe Ice King does have one Gunter he is particularly fond of. The Ice King uses Gunter as his personal servant, whereas the other penguins working for the Ice King are more or less slaves. It is not completely clear whether Gunter is male or female. However, in "The Chamber of Frozen Blades" Gunter lays an egg, which would likely mean he/she is female.

Gunter may be among the penguins seen in the Theme Song in the Ice Kingdom.

In "What is Life?," one of Gunter's known duties is being used as a stand for the Ice King's video game machine. Gunter also tried to warn the Ice King that Finn and Neptr were in the room.

Gunter is mentioned in "What Have You Done?" when Ice King says, "Gunter, dispose of Finn's carcass."

Gunter appears again in "It Came from the Nightosphere," where his/her soul is almost stolen and is dubbed "by far, the most evil thing I've encountered" by Marceline's dad, and thrown at the Ice King. The Ice King's response is "Gunter, who told you, you could fly?"

Gunter appears in "The Chamber of Frozen Blades" and is very sick, but it later turns out he/she was pregnant and the egg hatches revealing a floating pink kitten.

Gunter re-appears in the episode "Still" trying to mess with the potions Ice King brought to freeze Finn and Jake. Ice King says that Gunter likes to break bottles, which Gunter eventually does. He/she is shown to be a troublemaker in this episode.

Gunter appears multiple times in "Holly Jolly Secrets Part I" and "Holly Jolly Secrets Part II." Notably, during his "Penguin Dance Party" video, Ice King calls two different penguins "Gunter" in the same scene. It is unknown whether this was intentional by the show's staff, but this is the main in-show evidence suggesting that Gunter is not a specific penguin.

In "Five Short Graybles," Ice King makes Gunter float out to sea on an iceberg because he believes that Gunter is making his cave smell bad. It also shows that Gunter can play the keyboard, though not very well.

In "Princess Monster Wife," only one penguin is called Gunter and many other penguins are addressed with similar, though different, names.

In "King Worm " Gunter is part of a giant monster made out of penguins.

Gunter may also have appeared in other episodes featuring penguins, although it's impossible to tell by sight because Gunter is visually indistinguishable from other penguins.

Appearance
Gunter looks like a regular penguin with big black eyes. She/he has a black back, black, flipper-like wings, and a white flat belly. She/he also has a pointed yellow beak and yellow feet with three toes.

Relationships
The Ice King sometimes gets upset at Gunter, such as when he/she accidentally broke his video game or ate his socks, but the Ice King seems to appreciate his/her company. The Ice King often speaks to him/her like a father.

The episode "Princess Monster Wife" confirms that the Ice King indeed does know all of the penguins by name and all are named some variant of Gunter (e.g. Gundy, Gunder, Gunthy, Goonter, Günder (Gewn-der)...).

Trivia

 * Gunter is the name the Ice King gives to all of his penguin servants, yet he pronounces it differently depending on the penguin's personality. This is shown in "Princess Monster Wife."
 * Gunter is the most evil creature that Hunson Abadeer has ever seen, as he claims in "It Came from the Nightosphere."
 * In the episode "Holly Jolly Secrets Part II," BMO deciphers a complicated code in the Ice King's crying, which reveals an image of Gunter when printed.
 * Although the name is spelled "Gunter," it's been pronounced "Gunter," "Gunther," and "Goon-ter." It's unclear whether the varying pronunciations for the same name are intentional by the show's staff.
 * Gunter (variations: Günther, Gunthar) is an old German name made of the words Gund ("Fight," "Battle," "War") and Heri ("Crowd," "Folk") referencing the fact that Ice King calls all his penguins some variant of Gunter, making them his army.
 * Marceline's journal has the name Gunter on it in the episode "Marceline's Closet."
 * Gunter, according to the Ice king, likes to smash bottles and eat socks.