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troperville

Sam & Max
- Averted in the videogames though; Bosco, Stinky and Sal are neighbours of Sam and Max who are mentioned long before their on-screen introductions.

Literature

This happens in the Mary Minor, Mrs. Murphy and Tee Tucker mystery series written by Rita Mae Brown. The series is set in a small town in Virginia and there are always at least a few new characters introduced with each novel.
R L Stine's Ghosts of Fear Street series. It must be one long street...
- Some characters don't live on Fear Street, but if they don't, they do something that involves Fear Street or the also-cursed forest and lake beside it.

Live-Action TV

Early seasons of Smallville had a new meteor rock freak every week.
Everwood, including the flavor of the week patients.
British police show Heartbeat takes place in a small town of Ashfordly, which seems to have a ready supply of people committing various crimes, and in later seasons suffering from alcoholism and needing An Aesop.
Murder, She Wrote, complete with one of the new characters dying in the episode they are introduced. Seems like Cabot Cove is the murder capital of the east coast.
Cardale in Peak Practice, another medical show example. Possibly lampshaded in one episode:
"Who'd have thought we had a world class jazz musician in our town?"
The characters of Kate and her family in Robin Hood have apparently lived in Locksley all their lives, even though it's a reasonably small village and they've never been seen or referenced before Season Three.
The "small town" of Eureka keeps adding new characters to the point where Carter, who's been there approximately five years by now, still doesn't know everyone (or even, apparently, most people).
Midsomer Murders takes an entire county of small towns as its setting, but it's been running so long that this is a problem anyway.
- Justified, as nearly every episode has 3 or more deaths and without new neighbours, there would be hardly anyone left.
Buffy the Vampire Slayer and related media. Handwaved in that the Hellmouth the town is on attracts the monsters and scaries.
Star Trek: The Next Generation did this a few times, most notably in the episode "Lower Decks" which introduced a set of brand-new Starfleet officer characters ... only for them to never be seen nor referenced again. One of these one-off characters is shown to be Ten-Forward's secondary bartender (in lieu of Guinan) who is well-known enough to be a confidante, but had never been referenced before or since.

Video Games

In the first few days of playing Animal Crossing, a new neighbor moves in every day.
WarioWare series. Not explicitly stated how big Diamond City is as a place, but each game adds new characters apparently just living a few houses away from each other that have never been seen before in the entire series.
Arguably The Sims, in particularly The Sims 2 and 3 if you're following the stories of the Maxis/EA premade families.
The Reality On The Norm Shared Universe is set in a city which seems to gain more and more citizens with each installment.

Western Animation

Springfield started out this way, until all the side characters were cemented as a part of the Loads and Loads of Characters cast.
- Also, neighbors of the Simpsons (aside from the Flanders) occasionally appear at random, then disappear, never to be mentioned again.
  - Although this too has settled down a little. Flanders lives on one side, Laura Powers lived on the other side (until that house was bought by Sideshow Bob wearing someone else's face, with a brief mention of Powers leaving). More distant neighbors are seldom specified, although Mrs Glick seems to live on the same street somewhere and the house across the road was once owned by George Bush Senior and then by Gerald Ford.
  - That huge house opposite the Simpsons certainly seems to pop out of nowhere.
Fillmore! reuses the faces of previous episodes' characters, but they never speak or do anything of importance. Outside the safety patrol itself, no character features in more than one episode, even those who have a history with the regulars.