Weird Beard

Yarr! Davy Jones' beard be made of Octopus Tentacles, yarr!

So you want to give a character a beard and invoke all of those fun beard tropes. But oh no! Our character is some kind of creature or robot that doesn't have hair! But, that's okay, we'll just put something else there and make it look like a beard. See? All better!

May also be used when you want your character to appear alien, but not too alien.

Examples of Weird Beard include:

Anime and Manga

Dennou Coil had a whole episode about this. The beards were made out of "illegals", dark creatures that could only be seen by characters wearing the series' computer glasses. The beards worshiped the people whose faces they were on as gods.
In One Piece, Django's goatee is actually the stalk of a mushroom that grew on his chin. Eating the cap of the mushroom is what gave Django his hypnosis powers.

Comic Books

One of the mutated teens in Charles Burns graphic novel Black Hole has what appears to be a regular beard (which makes sense since he's been living out in the woods) but on closer and repeated viewing is revealed to be his mutation: a lump of flesh growing on his face in that general shape.
Empowered has the villain Weirdbeard. (It's made of Prehensile Hair, otherwise seems to be normal - just very long.)

Film

Davy Jones from Pirates of the Caribbean has Octopus tentacles around his face to resemble his beard in human form.
Sentinel Prime has weird circuit things that look like a beard in Transformers: Dark of the Moon.
- This is also true with Jetfire from Revenge of the Fallen.
The Zen Master (an old Japanese three-wheeled truck wearing a coolie hat and traditional Japanese robes and carrying a rake on one of his wheels who runs Tokyo Art Museum's zen gardens) from Cars 2 has a large white fender over his front wheel that looks like a long, flowing beard (appropriate since he's supposed to be a parody of various old men that coach martial artists).
- For some reason, his toy and the accompanying blister package art did not show him with his "beard", making him look much younger than he actually is. The toy also incorrectly referred him to as the "Zen Master Pitty", despite not being a forklift.
In The Hunger Games, Seneca Crane's beard is just... here.

Literature

In the Discworld novels, Jack Frost's hair and beard are made out of icicles.
Treebeard and most of the other Ents in The Lord of the Rings have moss or leaf beards.

Live Action TV

D'Argo from Farscape has a fleshy beard-bulb along with a long-braided moustache.

Music Videos

In the video for Lady Gaga's "Bad Romance", there's a guy with a beard made out of golden metal.

Tabletop Games

In Magic: The Gathering, male Kor have tentacle beards.
In Dungeons & Dragons, barbazu are devils with beards made up of bristles so sharp and stiff, they can stab people with them.

Toys

Turaga Vakama in the Bionicle movies had one on his mask that resembled Pharaoh beards. Other characters that shared his mask-type received noticeably shorter ones, even Macku, who's female. Also, Hahli's Toa Inika form had a mustache+goatee combo composed of breathing tubes. Again, Hahli's a she.

Video Games

Admiral Razorbeard from Rayman 2 has, well, a razorblade for a beard.
Emboar from Pokémon Black and White has a beard made of fire. Also, Probopass has a mustache made of iron filings and Beartic has a beard made of ice.
A minor character in the original Paper Mario, Oakley, has a beard made out of leaves.
The Dekus in The Legend of Zelda: Majora's Mask also have leafy beards.
In The Secret of Monkey Island, the Ghost Pirate LeChuck, on whom Davy Jones above is based, has a beard which appears to be made of hair but writhes about of its own accord. In the sequel, Guybrush keeps it as a memento - and it's still wriggling.
- Also, in The Curse of Monkey Island, the aforementioned Ghost Pirate LeChuck has a new demon form with a beard made of fire.
Much like Davy Jones pictured above, the male Draenei and Eredar of World of Warcraft have tentacle beards. (Or barbel beards, or something.)

Web Comics

In Questionable Content, the Linux AnthroPC has a heat sink where a beard should be.
Vlad in El Goonish Shive had feather-antennae growing out of his jaw and cheeks. He lost them when he was transformed into a human woman.

Western Animation

Flexo's metal beard in Futurama.
- Also Bender sports rust stubble if he goes too long without drinking.
Several Transformers characters have metal beards, goatees, or mustaches.
- In Transformers Generation 1 we have an apparent case of a metal beard growing. Alpha Trio has a long white metal beard, but during a time travel episode a younger version is seen with a black metal mustache. Best not to think about it.

Real Life

Egyptian Pharaohs wore strap on beards made of different materials. Even one, Hatshepsut, who was a woman.
There is an annual, international artistic contest, the World Beard and Moustache Championships. Just look at those pictures. One man who won Norway's Best Beard Contest scuplted part of his beard into a quite impressive deer. Another has a windmill made out of beard growing from his face.
Some people are into bee bearding, with several thousand of the live bugs looking like a beard on their faces and necks, although it's not likely to be a long-term thing when you think about it.
Dewlaps are fleshy dangling structures found on the necks of animals, such as turkeys (wattles), lizards, or zebu. Not technically beards, but it looks like this trope.
The barbels of many fish species can look like mustaches and goatees.