Permanent Elected Official

"Birch Barlow: You know, there are three things we're never going to get rid of here in Springfield. One: the bats in the public library. Two: Mrs. McFierly's compost heap. And three, our six-term mayor: the illiterate, tax-cheating, wife-swapping, pot-smoking, spendocrat "Diamond" Joe Quimby. Mayor Quimby: Hey! I am no longer illiterate!"

- "Sideshow Bob Roberts", The Simpsons

A politician with Ultimate Job Security. No matter how incompetent or corrupt, that person cannot be thrown out of office. Perhaps because there's no one to fire them (who fires the King of a Kingdom?), or perhaps because no one will run against them. Or maybe their corruption is part of the reason they can cling to their office. Lastly, they may simply use the population's general apathy.

In episodic media, this may overlap with Vetinari Job Security: the politician (more often incompetent than corrupt in those cases) does get replaced—and his replacement proves far worse, thus the heroes have to try and get the previous buffoon back into power.

Just the First Citizen is often when the dictator poses as this.

Anime and Manga

 * Mayor Papazoni of Pecola definitely fits into this mold. He is fond of giving long-winded speeches, but then simply pops into one of his many hidey-holes whenever somebody questions what he's going to actually do for the town. They state that the only reason that he keeps getting elected is because nobody else will take the job.

Literature

 * The character translated as Mayor in Gogol's The Inspector General, who is a powerful Corrupt Hick with more powers than a mayor would be thought of as having. This makes it Older Than Radio.
 * This was in Czarist Russia, there was neither a press nor was central bureaucracy as strong as today. Of course he had more power than a mayor of a comparable town today.
 * The Commdors of the Republic of Korell in Isaac Asimov's Foundation are a dynasty of these.
 * In David Weber's Honorverse, this happens with dismaying regularity. First, in the People's Republic of Haven, over half of the population is on state-funded welfare and the families of the "democratically elected leaders" have been in place for generations; they keep the population happy and they get to stay in power... even if they have to go conquering most of the other planets in the region and loot their economies to keep the edifice standing.
 * The Solarian League is worse, though; the Office Of Frontier Security is in bed with the transsteller corporations; the usual plan is to destabilize an independent star nation so that they'll "invite" in the Office Of Frontier Security for "protection"; the OFS guy is often "elected" to be "President For Life" while the transstellers take over the planet's economy and start looting.
 * Late Jim Cloop, the mayor of Scrote in Discworld, has the position permanently, because everyone agrees he's the best mayor they've ever had. Not "for life", permanently. He died in office immediately after the election, and they couldn't afford to hold another one. When his first term was up, they noticed he hadn't raised taxes, taken bribes or embezzled public funds, so they voted him in again. That was over a century ago, and he's still mayor as of the most recent Discworld Companion.

Live-Action TV

 * Boss Hogg in The Dukes of Hazzard is the County Commissioner, and completely corrupt. Nobody ever runs against him for commissioner, though. (Ditto for Sheriff Roscoe P. Coltrane.)
 * Probably helps that he owns the bank and everyone's mortgages.
 * Buffy the Vampire Slayer had Mayor Richard Wilkins III. He was credited as being the founder of the city of Sunnydale, and was still in office one hundred years later (having twice explained away his youth by adding a numeral and claiming to be his own child: he was also Richard Wilkins II and I). It was all part of his master plan, in which he'd also arranged for the graduation ceremonies of the town's high school to take place on the 100th anniversary of its founding, with himself as keynote speaker.
 * In Boardwalk Empire, Nucky Thompson seems to be this at first, but by the end of the season we see that his hold on power is much more fragile that it appears. The mayor and the Commodore both thought that they had achieved this status, but it was revoked when they lost Nucky's backing due to a changing political climate.
 * In Once Upon a Time, Regina Mills (formerly known as the Evil Queen) is always Mayor of Storybrooke, Maine, due to the terms of the curse placed upon its inhabitants. The citizenry rationalizes this by assuming that everyone is simply too afraid to run against her, even though no one actually seems to like her enough to vote for her.

Tabletop Games

 * In Rifts, there's Karl Prosek, of the Coalition States. He was elected Chairman after the death of the previous head, his father. He then secretly started a grass roots campaign that got him elected Emperor-for-Life. He's not incompetent, nor is he corrupt, but under his reign, the Coalition has become more and more intolerant of non-humans and magic users, and its populace has become more and more ignorant and xenophobic. In a way, he's actually worse than an outright tyrant, because he's convinced the citizens of the Coalition that his draconian measures are for their own protection, and they cheerfully obey his every wish.

Video Games

 * The SimCity series lampshades this by having your advisors comment on how corrupt you are, and some call elections a "formality" (given that the only way to be voted out of office is to work your city into inescapable levels of debt), and mention in passing "the debauched toga parties" and wasteful spending.
 * In the original game you would be outsted by "an angry mob, lead by your mother" if you fail to achieve the objetives in scenario mode.
 * In later SimCity games, you are known as "Mayor Defacto". They've come to terms with your eternal rule, and have just accepted it.
 * Monique Diamond, finance advisor from SimCity 4 is particularly egregious with implications of embezzlement. She's not complaining; it's also implied she's skimming money off the top herself.
 * It was possible to be fired (therefore losing the game) in the SNES version, though.
 * It is also possible to get fired in most versions of SimCity 2000 and 3000.
 * The Civilization games comes to mind... no matter how many centuries pass, no matter how many revolutions rage through the lands, YOU are always the guy who's in charge when the smoke clears - whether that means being the 'democratically-elected' President, Prime Minister, Glorious Dictator, or High Priest of the Church of You...
 * Ditto in Sid Meier's Alpha Centauri, where the leader of the faction is always the same. However, this is slightly justified in the Expanded Universe where the leaders undergo periodic life extension treatments. Also, unlike Civilization, the factions in SMAC are based in ideology, which would make sense that they'd want to keep the founders of the ideology in power. Also, in case of Yang, his State Sec makes sure he stays in power. Also subverted in an Expanded Universe novel, where Yang gets arrested and put in jail for his atrocities and his political rival Jin Long is put in his place... except Long is actually a clone of Yang and takes Yang's name for political reasons and marries Yang's daughter.
 * The DS games Drawn to Life seem to mistake a mayor for a king. As apparently when the mayor will eventually die, his daughter will then become mayor.
 * David Jefferson Adams in Shattered Union is very closely elected President for his first term, and gets all his opponent disqualified for the next election. His second term doesn't last long.
 * An ideal to be aspired to in Tropico, hard to achieve.
 * The Enclave in Fallout has such Presidents. There is President Richardson who was serving his sixth term by Fallout 2 and won otherwise democratic elections as there were no other candidates, and there is President Eden who proclaimed himself President after the formers death and has been the President for over thirty years.

Web Comics

 * Mayor Jeb from Men in Hats, a totally incompetent Corrupt Hick. As Aram puts it, "Why the hell do we keep voting for this guy?"

Western Animation

 * The eponymous mayor of Tom Goes to the Mayor, who has remained in office for twelve years due to a combination of family connections, low voter turnout, and a bizarre local law giving mayors thirteen-year terms.
 * Mayor Quimby of The Simpsons who tends to get voted back into office due to voter apathy, corruption, and the fact that the republican party can't pick a candidate to oppose him. When they do, they picked Sideshow Bob (a known criminal).
 * One of the future episodes revealed he was eventually indicted and reduced to working as a cab driver under the alias "Mohammed Jafar".
 * The Mayor of Dimmsdale an The Fairly OddParents once introduced himself as "mayor for life". When the crowd reacts with confusion, he merely laughs and tells them they need to pay attention to those ballots.
 * In the film version of Horton Hears a Who!, the Mayor of Whoville is revealed to be an inherited position, which implies this. Of course, Mayor seems to be just ceremonial, as the City Council seems to hold all the political power.
 * The Powerpuff Girls: The Mayor of Townsville is a completely incompetent moron but manages to stay in office. One episode revealed that this is because no one ever runs against him - though he insists on campaigning anyway which annoys everyone so much that when Fuzzy Lumpkins tells him to shut up, the people vote him the new mayor despite the fact that he never actually enters the mayoral race. Of course Fuzzy turns out to be even worse so the girls get the old Mayor to take his job back.