Mention the most unique song that you have ever listened to.CHOPSTICKS the happy song https://youtu.be/ojNl-xJF_2w
At the moment the one that comes to mind is “Lizard” by King Crimson ( remastered by Steven Wilson of Porcupine Tree fame and various other accolades) which is actually a suite and somehow manages to fuse a little bit of everything including rock, free jazz, and both modern and symphonic classical into one composition with very satisfying results but it may take several listens to soak it all in. Plus a very special guest vocal that kicks things off. Have had trouble getting it out of my mind lately.
Pierrot Lunaire from Arnold Schoenberg, it can come as quite a shock the first time but maybe it doesn’t count being a Sprechgesang.
Jackyl -- "The Lumberjack"

Gotta love a song where the lead instrument is a chain saw.
Andrew Gold - from the Album "Fraternal Order of the All" - Ccosmicc Ccarnivall. The album is fantastic - some really melodic tunes - but this song is off the wall.
Pink Floyd....Several Species of Small Furry Animals Gathered Together in a Cave and Grooving with a Pict
Something can be unique, that renders most unique redundant.

That said, I nominate Hocus Pocus by Focus
White Bird - It’s a Beautiful Day
21st Century Schizoid Man - King Crimson
Interstellar Overdrive - P Floyd

unique has many meanings....

San Franciscan Nights by Eric Burdon and The Animals....

or Pineapple Pen.....
I would second the person who said “Birthday” The Sugar Cubes.  Bjork should be high on a list of unusual songs. 
Wow, where do I start. Well, any Yoko Ono song is on the list, if your hearing works after.   But for real George Benson's White Rabbit is wonderful. 

The Necromancer...... RUSH

To name just one.

"Unique songs" is kinda what they do
 "Irreversible Neural Damage" track on Kevin Ayers "The Confessions of Doctor Dream and Other Stories" album. A rather creepy number with vocals by Kevin and Nico.
’Fish Heads‘ by Bill Mumy of ’Lost in Space‘ fame is hard to top.


Childhood stardom does different things to people........
Last Plane Out, Toy Matinee

Industrial Disease, Dire Straits

Head Like A Hole, NIN

Cuts You Up, Peter Murphy
I was about 14 and my brother brought home an Elton John album before any of his music was on the radio. I had never heard of him. “Where To Now St. Peter”
i remember thinking, “Who is this British guy, and boy can he play the piano!”  For that window in time, he seemed unique to me. Obviously, unique is in the eye of the beholder, and one’s opinions change with time. Similarly, all of us can think of novels that elicited a very strong reaction within us, and when we try and re-read them years later, some of them do not age very well. I still like the EJ song!
Maori, wedding chant.. Very emotional. Everyone in the party began the Huca Huca Huca, it slowly became more and more intense. Then a climatic HUCA, HUCA, HUCA and STOP..... I’ve never seen anything like it.. I was very honored... They became one family from two... A thousand years old tradition...

BTW no pictures... steal your spirit... no jokin’ matter, wind up "part of" the wedding feast.. Yup they use to eat folks... Long Pig...

All songs are unique, or should be.  Except those that are plagiarised, which is many.

By the way, 'most unique' is not a useable expression.  'Unique' means that there is no other item the same.  A thing is either unique or it is not.

OP should have written 'most unusual' or suchlike.
Chantilly Lace by the Big Bopper....First lyrics that I could understand as a small kid, made me smile!
Forgot all about Napoleon the 13. I'll add Hitchhiker Joe by The Rugburns