I also could never pick just one. But Tears of Rage by the Band would be way at the top of my list.
Labdad Questionmark and the mysterians.
When a man loves a woman…by Percy Sledge ,has got to be one of the best of the 60ies along with Leah by Roy Orbison.
duh I just noticed your ? mark labdad and 96 tears is still one of the greatest!! oh yeah…Lion sleeps tonight by the tokens
The Doors: Light My Fire
The Animals: House of the Rising Sun
Loving Spoonfull: Daydream
The Zombies: She’s not There
The Searchers: Needles and Pins
And lots more
wow, i always thought “I’d love to change the world” sounded more like an early 70’s burnout song than an earnest 60s…come to think of it, Allmusic.com says it was 1971…i rawk 
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My fave? any of Pink Floyd’s early recordings, I can’t pick just one…its a choice between “See Emily Play”, “Lucifer Sam”, and “Apples and Oranges”.
- apologies to those who consider the early 70’s to be the 60s. I do too, but I think we are talking chronological in this thread.
Grateful Dead: Dark Star (from the “Live/Dead” album)
Bob Dylan: Mr. Tambourine Man
Phil Ochs: The Crucifixion
Janis Joplin: Me and Bobby McGee
Jefferson Airplane: Volunteers of America
Jimi Hendrix: Are You Experienced? (also his version of the US National Anthem)
Arlo Guthrie: Alice’s Restaurant
Country Joe and the Fish: Fish Cheer->Feel Like I’m Fixin’ To Die
That’s just a start, since I really like the music of the hippie/folk/Vietnam, Civil Rights and etc. era.
C’mon, people, you can only have one favorite. I know it’s hard, but if I had to choose one, it’d probably be “Up on Cripple Crrek,” by The Band. Of course, if you ask me tomorrow I’ll probably give you a different answer.
Knock on Wood by Eddie Floyd. Greatest Memphis R&B song ever, and one of the high points of western civilization.
Foxy Lady by Hendrix. Love to crank it up for the screamin feedback explosive intro.
Underappreciated one-hit wonder: Concrete & Clay by Unit Four + 2 or something dumb like that. Catchy.
Underappreciated multi-hit wonder: Have You Seen Your Mother, Baby, Standing in the Shadow by the Stones. Never hear it on the radio.
Venus in Furs by the Velvet Underground. I was convinced for a long time that that song was done in the mid to late 70s, it was so advanced for its time. When I found out that it had been done in 1967, my immediate reaction was - “Why was anyone listening to Sgt Peppers’, then?”
That Liverpudlian album of novelty songs still gets more credit than the Velvets do for changing the face of music, unfortunately.
I’m always torn how to write the name of the group. The label on my 45 has the group designated with the punctuation mark, “? and the Mysterians”. But my album shows the group as “Question Mark and the Mysterians.” I tend to favor “?” though, because Rudy Martinez had his named legally changed to the punctuation mark.
Jeez, could I get any geekier about this? 
So many excellent answers, so many more to choose from. I’m kinda partial to:
Procul Harum - A Whiter Shade of Pale
Stones - Paint it Black (as mentioned above by Gyan9)
Beatles - Nowhere Man or Eleanor Rigby
And, Exapno, it was a great time to be young and listening to the radio. All the time.
I don’t think I can limit myself to just one song. My top three for the moment includes:
- Sam Cooke “A Change is Gonna Come”
- The Doors “Five to One”
- Otis Redding "Respect
I reserve the right to change my mind at any time, and I probably will.
“My Little Red Book,” by Love.
Build Me Up Buttercup - The Foundations