I'm currently playing two -

First is Jonsi, the 88 year old Dwarf Cleric who just decided to go adventuring. No tragic history - his parents and brother are still alive, no horrific destruction of his home town... He just caught the Viking wanderlust. He's got a good sense of humor, and is happy to drink and tell tall tales.

The other is a 14 year old female Paladin / Warlock, named Electra. She found an ancient looking sword in a lake, but when she holds it, it gleams and is razor sharp. This "illusion" transfers to her as well - she can appear older, taller, or as a boy. She has used that to draw less attention to herself, since a small 14 year old girl is a little odd in this rough world. She can't quite control it - sometimes she's not entirely sure which "persona" she's going to appear as.

My Tiefling Bard, Bellista, is 18, and has been through some sh*t.

I once played a swarm of rats nature cleric. It was odd.

My DM and I ended up creating a 1/2 celestial monk in 3.5e that was the earthly form of a Solar trapped in a human body.  He had disobeyed orders to try to kill the BBEG lich with a group of adventures and got punished when everyone died.  We decided he had a lot of limitations and was crazy because of the expanded mind being crammed into a human body.  I made sure it was only humorous things and he was never annoying.

Some of the weird things he did.

1. When soldiers tried to confiscate the party's horses while he was hidden on top of a thatched hut: he took burlap sack filled it with thatch, drew a horse face on it, and put it on his staff.  He then tried to give the hobby horse to the soldiers.  The soldiers had no idea what do do and we kept our horses.  After that the monk rode the hobby horse everywhere with a speed of 50+.
2. Convinced another character that licking rocks caused them to glow as he cast Daylight on them.
3. Had a robe of useful items and made everyone close their eyes when he got something out of his "backpack".  He made horrible pained noises and would never tell where the items came from.  Once it was a 10 foot pole.
4. When meet by the party in a dungeon that had no way in except from the air, he claimed he tripped.  This caused the party to not trust him at all.  Of course he had flown in with his wings that would disappear once they have been used for 1 hour.
5. When the leader of the party was killed by an assassin he transformed into a Solar in a blaze of holy light, cast resurrection during the blaze, and transformed back before the blaze disappeared.  When the light came back he was performing the Heimlich maneuver on the leader and claimed he just had swallowed something wrong.

I had a number of other comical things planned, but the group broke up and we never got back to it.

That's great

Last year I ran a chaotic good perky goth Drow necromancer.  Her favorite things were sunshine, flowers, and puppies, and she was a total daddy's girl.  Basically I tried to make her the most un-drowlike drow I could think of.

Ok, now my weirdest one has got to be a character for a mini campaign. She is a teifling love (sorry, unity) cleric of sune, who has no memories before 4 years ago. I’ll post her story in the character backstory page. She plays like a stereotypical bard, and it is cannon that she has protection from pregnancy from sune. This is going to be......... interesting.

Another of my unused ideas: A sorcerer who is unaware they have magic, and whenever they use it they think it's just a crazy coincidence that that thing happened

A Drow Warlock, The Noble Genie Patron, who wants to be a normal elf because they are nice.

Update me on their adventures. I must know more.

I play a beasthide shifter dragon sorcerer (black dragon) multi classed as a  fiend pact warlock (baphomet) who was abandoned by his clan at birth due to his scales he was thought a freak and so now he believes himself a monster and will do anything to become more monstrous. Because of his pact he now has a wolfish snout and bull like horns as well when he shifts he grows to 8ft tall and takes on a look more like a were wolf meets were bear with bull horns and luminous green eyes

I haven't tried this yet, but at some point I want to run a Circle of the Moon Warforged Druid.

Because there are days when you need to make all the Beast Wars jokes.

Genius 

(i wish i got that joke) 

Ah yes, the old Transformers series that made literally no sense.

Quote from Sly_Fox >>

Genius 

(i wish i got that joke) 

 Transformers: Beast Wars.  It was a late 90s cartoon and toyline where the characters turned into animals rather than vehicles like most Transformers.

The original cartoon was way better.

I kinda love this about the normal upbringing Rogue.

My weirdest was named Timoden. He started as a character from a homebrew Pathfinder campaign. In 5e he became a variant human Viking cleric of Thor (Life cleric which isn't exactly RAW). His first adventure was Harried in Hillsfar (possible spoilers) and since Goats are the sacred animal of Thor this was the best adventure possible for him. He adopted the mother goat Jovee and started a goat farm where he bred a bunch of demonic looking goats from her. He saw the goats as a divine blessing from his God so even as a Cleric he had no moral qualms about doing so. The demonic appearance was just cosmetic anyway and statwise they were normal 1gp goats. I spent all my downtime on goat farming even when it left me at level level 4 when everyone else used downtime to get to tier 2 faster. 

Boronith Kazequeonde, A Goliath War Cleric,

A delusional maniac who serves Bane, the God of Tyranny, But Boronith believes that Bane is the God of destruction, and will do anything Bane asks, even if it is the the detriment of his party.

That sounds like a character concept that would get old quickly.

I ran a campaign where someone actually *did* play a Warforged Circle of the Moon druid. Yes, we did make Beast Wars jokes. He also ate the body of one of the big bads after the party killed him. So there you go.