"Taxman" (The Beatles)

Although "Taxman" from Revolver (1966) wasn't George Harrison's first foray into writing songs, I think it is the song where he started to step out of Lennon and McCartney's shadow.  There's a real sense of confidence and bile in it: Harrison was pissed off about being in an income bracket in the UK where the majority of his earnings were going straight to taxes, and he let fly with this terse, scorching editorial.
You can't completely count out the other Beatles' contributions, though—particularly McCartney.  Macca does double duty on the track, cranking out the hellacious, Stax-indebted bassline (the man sounds like he has four calloused thumbs on the section from 0:55 to 1:06), as well as the raunchy, Eastern-tinged lead guitar solos at 1:12 and 2:25.
For me, it's the best Beatles album opener, next to "Come Together."

"Tomorrow Never Knows" (The Beatles)

I remember watching the Beatles Anthology on TV back in 1995 and hanging on every word that George Martin and the surviving Beatles said regarding the creation of "Tomorrow Never Knows" from the 1966 album Revolver.  
(Of any song from the psychedelic era, it really holds up to repeated plays, even now.  I'll even go so far as to say that its innovative use of loops and rhythm were a direct precursor to hip-hop and trip-hop.)
According to George Harrison in the interview, John Lennon had read a copy of Dr. Timothy Leary's book The Psychedelic Experience, which is basically a how-to guide for tripping, based on the Tibetan Book of the Dead.  Apparently the notions of "turning off one's mind and floating downstream" came from the book and made their way into Lennon's lyrics.  
Lennon also envisioned the music having a transcendental vibe, with the entire song remaining on a single chord, much like an Indian raga, and his voice being treated in such a way to make it seem as if he were singing from a mountaintop.
On each take of the song, the boys lock in on a C chord.  Likewise, Lennon's vocals are sent swirling through a Leslie speaker to give them that far off effect.
But everything else about Take 1 and the final take that appears on Revolver are vastly different.  The original take is like a funeral dirge with this murky, underwater feel and Ringo Starr drumming in this march pattern, not the hypnotic groove everyone is familiar with.
As for the ambient noises on the final track, Paul McCartney explained that these were tape loops that he'd chopped up/spliced at home and brought to the studio in a plastic bag.  (For instance, the famous "seagull loop" heard on the song is actually a recording of McCartney laughing, sped up.)  Being that these were actual loops of recorded tape, they had to be spooled onto machines throughout the studio for playback, with people holding pencils, jars, their fingers, etc. in place so that the loops would spool properly and play at the desired speed.  Then the loops were fed into the mixing board, and everything was mixed live--meaning each sound on the final recording is the result of The Beatles pushing the faders spontaneously.
As a music geek, it just gives me chills thinking about the unbridled sense of creativity and collaboration in the studio that gave rise to this track.