In Gordy the head of a major corporation hands over control to the title character, a small pig, upon his death. The film treats this as a good thing.
A rare case of such a character being the main protagonist, the film Baileys Billions had the titular character, a dog named Bailey, inheriting $1 billion from a deceased relative, causing the woman's nephew and his wife to try to kidnap the dog to gain the inheritance.
Similarly, the movie The Aristocats had a Paris woman deciding to allow the Cats to inherit her estate, with her butler intending to off the cats before she dies so he could acquire the inheritance.
And, ironically enough, that turned out to be completely pointless: if he had listened to the whole will, he would've heard that upon the Paris woman's death, the butler was to be appointed the cats' caretaker, so he actually would've (sort of) gotten the money anyway if he'd just left the cats alone.
A past Patrician of Ankh-Morpork (the aptly named 'Mad Lord Snapcase', though most Patricians had names like that) appointed his horse as a city councilor, though it is pointed out that it wasn't a bad councilor compared to some of the others, which included: a vase, a heap of sand, and three people who had been beheaded.
In Making Money, when Topsy Lavish dies, she leaves 50% of the shares in the bank to her dog, Mr Fusspot (who already owns 1% of the bank, giving him a majority and making him chairman). She then left the dog to Moist von Lipwig, forcing him to take control of the bank, which was her plan all along.
The Discworld Companion states that the Sto Plains village of Scrote has elected a dead body as its Mayor for several decades. The first time it was because the candidate died in mid-election (but was still more popular than the alternative); after that, the villagers really liked how the late Mayor hadn't raised taxes or embezzled town funds.
There is also the (in-universe) historical example of St Ossory's ass (his donkey, not his behind) which was made a bishop in the Omnian church.
Lord Midnight in the backstory of the Vorkosigan Saga, a horse who gets appointed heir by a Count as part of a feud between the Count and his son. The funniest thing about Lord Midnight is that he's important. When his appointment was legally challenged, it was upheld— thus establishing the precedent that a Count's designated heir need not be his own descendant, or even human. This becomes important when cloning and genetic engineering technology starts to blur the traditional definitions of what makes someone a descendant, or a person.
Midnight fortunately/sadly predeceased the Count, who designated his now reconciled son as heir once again.
One Andermani Emperor tried to do this for his potted plants in the backstory of the Honor Harrington novels. He was quietly deposed.
One of the many signs that the Helmacrons from Animorphs are completely insane is that all of their leaders (or ship captains at least) are examples of this trope, being corpses. They believe that the best leaders never make mistakes. Since the living inevitably make mistakes, dead people are the best leaders. Of course being "promoted" this way doesn't really hurt them at all since all Helmacrons have Born-Again Immortality.
Discussed in Lord Chesterfield's Letters to His Son. "Henote the Cardinal de Retz is persuaded that when Caligula made his horse a Consul, the people of Rome, at that time, were not greatly surprised at it, having necessarily been in some degree prepared for it, by an insensible gradation of extravagances from the same quarter." (letter 50)
In Malcolm in the Middle, when the scandals at Hal's company finally come to light one of the many revelations is that the chief financial officer had his dog on the board of directors. Hal himself is disturbed that he had C Cd the dog on several memos.
On 30 Rock, Don Geiss stacks the board of directors with "the most reliable collection of sycophantic yes-men this side of an Al Franken book signing: His golf cronies, his army buddies, a collection of unemployable family members and his hunting dogs." It's unclear how intelligent the dogs are, but they're apparently capable of staying seated at the table and voting in some fashion.
Referenced in The Suite Life on Deck, when Bailey explains her idea of a "big city:" "Any place where the mayor isn't a goose."
In one episode of The Fresh Prince of Bel-Air, there is a rather loopy corporate executive who took investment advice from his dog (There was mention of him investing in General Motors because the dog chased a Buick). By the time of the episode, Sparky the dog had been dead for months - but his owner had him stuffed and mounted in his office and still took investment advice from it.
In Fossil Fighters: Champions, the player and his friends are tasked at one point with finding a princess who has entered the tournament in secret, being shown a picture of a girl and her dog. The punchline to this sequence is that the princess is the dog (the girl's her retainer). This is lampshaded as absurd, but it apparently works for the country.
The president of the Henchmen Guild in Nodwick is a hamster.
Of course, the hamster only got appointed to membership in the Guild by its owner, the previous president. It was elected president at the next election due to all people present and voting being drunk. The hamster's reelection (for life) was due to the guild steward being bribed.
King Steve of 8-Bit Theater has a coffee stain named Rodney as his right hand man. He also briefly lost his kingship to a length of string.
Before joining Sturgeon Law, Pierce was senior vice president under such a CEO.
In The Simpsons, the nuclear plant is actually owned by a canary, so that if the government ever conducts a proper investigation, the canary is the one that will go to jail. Mr. Burns implies that this is standard practice, saying "Standard Oil was once owned by a half-eaten lunch." In another episode, he made a dog a vice president.
In another episode, an inanimate carbon rod from the nuclear reactor was honored as Employee of the Month (much to Homer's chagrin), and after Homer used a different Inanimate Carbon Rod to lock the door on the spacecraft after it was destroyed, it was given its own ticker-tape parade.
In fact, in the aforementioned episode with the canary, Homer is shown at the very bottom of the nuclear plant's employee hierarchy via a chart (the canary being at top, with Burns directly below him). Directly above Homer is the Inanimate Carbon Rod.
Its also implied that hiring animals to even work at the plant, never mind making them high officials, was one of Mr. Burns' quirks, as in the episode "The Last Temptation of Homer Simpson", it was shown that he had, aside from hiring an illegal Iranian immigrant worker, also had hired a duck named "Stewart" as a low level employee who tows nuclear waste. (He also berates it to "get back to work!" when it briefly stops to catch it's breath) This may be less outright insanity and more Mr. Burns' cheapness combined with a callous disregard for life. Later in the episode Burns is scanning the security monitors trying to find a pair of employees who are not currently fighting with each other to send as his representatives to a convention, not only is the duck seen fighting another employee, he's winning.
In the episode with the flashback of Maggie's birth, Homer briefly quit his job. When he returned to the plant he found he had been replaced with a chicken. Which he later ate.
Yet another episode had most of the Springfield Republican Party mistakenly assume the mayoral candidate they've chosen to back was a water cooler. They seemed to think it would make a fine representative.
On Gravity Falls, it is discovered that the real town founder was Quentin Tremblay, a Cloud Cuckoolander who was once elected President. Among his acts during his short term, he appointed babies to the Supreme Court.
The South Park episode "Douche and Turd" had a camp of PETA members who had a goat (one that was a doctor, apparently) as their leader.
In American Dad!, the secretary of Steve and Roger's sort of real detective agency is a stuffed bear named Teddy Bonkers, something Steve eventually gets fed up with.
"Oh, for the love of God, Roger! Bonkers is not real! Bonkers is a stuffed bear, a toy! I didn't want to tell you, but I hate this bit! Here's why, it's stupid! A teddy bear is our receptionist? It doesn't even make sense! Do you know how many calls we've missed?!"
One Bugs Bunny cartoon had Bugs and Yosemite Sam running against each other for mayor. They were defeated by a dark horse candidate. An actual dark horse is the town's new mayor. (Mare?) Bugs doesn't count, since he's a Talking Animal. The new mayor only neighs.
In Wander over Yonder, Wander finds a creature who, by accident (long story), he calls Captain Tim. Wander subsequently gives Captain Tim over to Lord Hater as a pet. Hater makes "Captain" the creature's actual military rank and thus outranks most, if not all of his minions.
Trope Namer: Emperor Caligula famously planned to appoint his favorite horse Incitatus to the consulship, the Prime Minister of Rome. However, it's a little more complicated than it sounds. He may have done this in order to mock the Senate. ("My horse could do as good a job as any of you clowns! I'll just teach him to stamp once for 'yes' and twice for 'no comment'.") It may have been a rumor started after his death. Or he genuinely might have been completely insane, which was the firm position of most ancient writers.
Mind you, most Roman historians were of the Senatorial class. If Caligula was deliberately and clearly insulting the Senate, they might have taken their revenge in the history books.
Real Life examples happen all the time (especially in small towns in the United States) with towns electing animals, and in one unconfirmed case of a statue, as mayors for their cities.
Although not officially, one of the things that the seventies era Feminist movement did was crown a live sheep Miss America on a boardwalk of Atlantic City in 1968.
It is common practice that the mascots of military units around the world (though mostly in the UK) hold pretty high military ranks. Nils Olav, the mascot of the Royal Norwegian Guard, is a King Penguin from Edinburgh Zoo, but not only does he hold a fricking knighthood, but also the official rank of Colonel-In-Chief. Oh, and just to clarify, he is thus technically closer to the Norwegian King than the actual Guard Commander, a Lieutenant colonel.