Done in an old Chuck E. Cheese commercial. The voiceover says, "Boring restaurants make you wait...and wait...and wait," while time lapses are shown with a kid sitting at a table waiting, ending with cobwebs and a ridiculously long beard. This is contrasted with the kid excitedly playing games at Chuck E Cheese.
Fist of the North Star: Not only does Kenshiro grow a beard in the animated movie in between his assault by Shin and encountering Bat and Lin (And then shaves it using some thug's knife), but he also develops a beard at the start of the second series, shaving it once he finally begins to fight again.
Done in Thorgal - after he escapes from caves where time flow was slowed, he wakes up with a long beard the next day. Not at all played for comedy, as the girl he was with had spent most of her life there and got aged into a shrivelled husk overnight.
Superman in the Time And Time Again story arc in 1991 gets a Time Passage Beard as he is stuck in the prehistoric past with no way of getting home except to expose himself to explosions and powerful energy discharges.
In The Kingdom of Gods, Sieh keeps magically aging whenever he uses too much of his power or does anything particularly unchildlike, such as remembering he's a father. Each time this happens, he ends up with a massive beard and head of hair.
The artificially aged Fourth Doctor in "The Leisure Hive" grows an incredibly long beard as the result of the process.
In the sixth series, the opening two-parter has the Doctor (who had always been beardless previously) imprisoned for several months and growing a beard. It happens again in the season finale.
Also from the revived Doctor Who; in "The Doctor's Wife", Amy stumbles across Rory's dead body in a Room Full of Crazy, having grown a beard, aged, and eventually starved to death. Don't worry, he gets better.
House the season 2 finale has House awaken from a coma. He determines roughly how long he had been out (three days) by the amount of growth of his Perma Stubble.
Chief O'Brien grows one in the virtual prison cell that he spends twenty virtual years in in an episode of Star Trek: Deep Space Nine.
In How I Met Your Mother, Ted becomes part of a very long discussion about Teddy Roosevelt with another man and a woman. He explains it went of for "days" and by the end, the other man and Ted have grown long beards.
At the end of 24's fifth season Jack is taken prisoner by the Chinese government and held there for nearly two years. When he's released in the following season he how has a long, unkempt beard as well as shaggy hair, similar to the James Bond example.
Sherlock in the Series 3 opening of Sherlock, during which he'd been kept a prisoner in Eastern Europe. He has it all shaved off by a barber upon his rescue. (This is used as symbolic contrast to the moustache that John grows.)
In one scene in the "Blood Gulch Chronicles" of Red vs. Blue, Church spends a thousand years standing in a hallway waiting for a computer to fabricate a teleporter (...maybe). By the end of it, Church has a massive beard drawn on his sprite. Played for laughs, given he's in a robotic body at this point.
Fire Emblem: The Blazing Sword has a short scene in the epilogue where Eliwood and Hector meet up several years after the events of the main plot, with Hector having grown one of these. By The Sword of Seals, which takes place over a decade later, it's grown even longer.
Don't Starve has this as a game mechanic with the character Wilson. While he isn't the only character with a beard, he's the only one capable of growing it over time and shaving it to obtain beard hair to use for dark magic.
Inverted by Roy on The Order of the Stick: after the group has been trapped in a Lotus-Eater Machine for an undetermined amount of time, he figures it couldn't have been too long since he doesn't have to shave.
Parodied in an episode of Tiny Toons where Babs is looking for a personal hero in old cartoons. As she sits in the Acme Looniversity archive the camera pans around and when it comes back to her she has a long gray beard and a cane. She tosses the cane aside, then rips the beard off and is about to storm off in frustration but decides to watch one more film.
Averted in The Smurfs episode "Smurf Van Winkle" (and its comic book counterpart "The Strange Awakening Of Lazy Smurf"), where the only characters seen with any Time Passage Beards are the Smurfs that claim that Lazy has slept for a few hundred years. Of course, the whole thing was a Faked Rip Van Winkle, since the Smurfs were wearing fake beards.