"All Along the Watchtower" (Jimi Hendrix)

"All Along the Watchtower" is probably one of Bob Dylan's creepiest compositions.  It's involves Pinter-esque dialogue between two outcasts (the joker and the thief), set against a backdrop of apocalyptic images: howling winds, wildcats growling, and general upheaval.  At the end of the song, you're left with the feeling that the clueless patricians and their servants in the castle are about to get a rude awakening from these "two riders."
Dylan's own version on his album John Wesley Harding (1967) is good and spare.  
But Jimi Hendrix's dramatic rendition makes it easy to forget that Dylan's version existed first.  In fact, it kind of makes you forget that Hendrix didn't pen the song himself.
According to Chris Salewicz's liner notes for the now out-of-print compilation Jimi Hendrix: The Ultimate Experience (1993), Hendrix heard Dylan's version while at a party with guitarist Dave Mason of the band Traffic.  As he and Mason discussed the song, Hendrix decided on the spot that he had to record it, although he was extremely nervous about being able to do the song justice—which explains why, even though he recorded his version in January 1968, it didn't see the light of day as a single until October of that year.
Hendrix pulls out all the stops: there's layer upon layer of electric guitar—sometimes fed through a Leslie speaker, sometimes a wah-wah pedal, and other times panned between speakers through production wizardry; every solo is tasteful, strategic, and used to maximum effect.  He also plays a mean bass on the track (I actually prefer his nimble bass playing to Noel Redding's plodding style), keeping metronome time with Dave Mason, who's strumming and slashing away on the acoustic guitar.  As always, Mitch Mitchell's drums I could take or leave.  Too many sloppy fills, not enough groove.  But, thankfully, it doesn't hurt the track.
"Watchtower" (and the 1968 album Electric Ladyland on which it appears) solidified that there was more to Hendrix than electric blooze, pyrotechnic stage theatrics, and psychedelia-tinged 3-minute singles.  It cinched his lasting reputation as a priest of sound.

"Just Like a Woman" (Bob Dylan)

I'm a fan of Bob Dylan's music and words but not always a fan of his recordings.  (That's why I usually prefer other artists covering his stuff.)  Too often, it's painfully obvious that he and his backing musicians ran through a song once, rolled a joint, and then pressed "record" on run-through #2.  Some might call it tossed-off charm or warts-'n-all immediacy.  But I call it sloppy.
Don't get me wrong.  I wouldn't expect a Dylan album from the mid-60s to sound polished and glossy—that's not who he was/is.  But hearing a brilliant song like "Visions of Johanna" get marred by musicians fumbling their way through chord changes, quite frankly, sucks.
One of the reasons I like "Just Like a Woman" from the 1966 album Blonde on Blonde is because Dylan and his band sound rehearsed—not to the point of sounding rigid, but simply ready to record: they know what the changes are, where the bridge is, where Dylan is going to stop and start, etc.
Anyway, the song is rumored to be about Edie Sedgwick, a wealthy NYC socialite and member of Andy Warhol's "Factory" crowd, who had a tumultuous relationship with a friend of Dylan's.  To deal with the rocky relationship and her party lifestyle, Sedgwick turned to booze and pills as a crutch—vices that ultimately led to her accidental death in 1971 after various stints of trying to get clean.
In a way, it seems plausible that the lyrics are a critique of a high-fashion debutante who's into staying high to avoid reality.
But I think that's too easy.  Dylan was never that simplistic.  Or quite that misogynistic.
My take is that the "woman" in the chorus isn't specifically Sedgwick but more of a metaphor for dependency and vice in general.  In other words, the amphetamines (and whatever else) seem to provide the kind of unconditional love, acceptance, and shelter that one craves from another human being; however, by their very nature, they are ephemeral and take more than they give, leaving the addict wounded and broken in the end.
In any case, it's one of his best ballads and best-sounding recordings of this period.

"Positively 4th Street" (Bob Dylan)

There's still a lot of speculation about what/who "Positively 4th Street" is all about.  I think it's pretty obvious that it was Bob Dylan's rebuttal to his detractors for "going electric."

Let's check out the timeline: Dylan started toying with mixing folk and rock in 1964, moving away from the acoustic protest songs, like "Blowin' in the Wind," that had earned him renown and respect in the folk music community.  The initial product of his folk-rock experiments was the half electric, half acoustic March 1965 album Bringing It All Back Home, which garnered mixed reactions among his folk music fans.  Not long after, Dylan released the sprawling rock single "Like A Rolling Stone" on July 20.  Five days after the release of "Like A Rolling Stone," Dylan played the Newport Folk Festival in Newport, RI, on July 25, where he decided at the last moment to play an electric set.  

And he got booed.  

Some Dylan contemporaries and fans postulate it's because his set was too short or the sound quality was poor.  Others (including Dylan) contend that it was because his new, electric sound had pissed off traditionalists in the folk community.  Whatever the case, the crowd reaction cut Dylan to the quick: the very people who were so ready to claim him as the folk movement's poster child one moment had suddenly and nastily turned on him the next (as you can see in this trailer for the 2007 documentary The Other Side of the Mirror).

His very next single was the acerbic "Positively 4th Street," released September 6, 1965.

Musically, the song floats along on a sublime, upbeat melody with bright splashes of organ and slightly-distorted rhythm guitar, courtesy of Al Kooper and Mike Bloomfield, who'd both performed with Dylan at Newport.

Lyrically, it's napalm.  The words come off as if he's berating a single individual, and—in a sense—he is.  The subject of his derision is symbolically "4th Street," which most certainly refers to West 4th Street in New York's Greenwich Village—the geographic center of the folk movement where Dylan got his start.  His vitriol is aimed squarely at the community he feels stabbed him in the back after he "went electric."  He cries foul that "folkies" were genuinely hurt and disillusioned by his stylistic shift, telling them You have no faith to lose, and you know it.  He also points out that the community still wants to claim him as its own, using his success for its own ends, but then never hesitates to disparage his mainstream acceptance.  You just want to be on the side that's winning...
It's Dylan at his thrifty, snarky, direct best.

"I Shall Be Released" (The Band)

The previous entry, Etta James's "I'd Rather Go Blind," was co-written by Etta James and her musician friend Ellington Jordan, who was serving time in prison and pining for freedom.
In a similar vein, "I Shall Be Released" is a Bob Dylan composition, written from the perspective of someone who is in prison, reflecting upon his incarceration and waiting for his release.
At least, on the surface "I Shall Be Released" is a song about a prisoner counting the days until he regains his freedom.  Another read of the lyrics reveals a deeper examination of the human condition and mortality, where the "prison" is our earthly body and the ultimate release is death.
In addition to Dylan's masterful lyrics, The Band's masterful interpretation of the song is the reason it appears on this list.  Ultimately, the song is more than the sum of its parts--just like The Band itself.  There's Richard Manuel's falsetto lead vocal and heavily reverbed piano that both float, weightlessly, above the song.  
Then, there's Levon Helm's percussion.  I'd always assumed that he had somehow created that slow march rhythm by using brushes against his snare drum.  Years later, I discovered he got that sound by strumming the wires on the back of his snare, almost like a banjo!
Next, Rick Danko's bassline has a melodicism and sense of movement that is one part Beethoven and one part James Jamerson of Motown's Funk Brothers.  
But my absolute favorite part of the song is Garth Hudson's wah effect on his Lowrey organ.  It is so subtle that it's almost easy to miss or mistake for some kind of envelope filter on a guitar.  (The only guitar on the track is Robbie Robertson's straight-ahead strumming in the right channel.)  Hudson creates this ethereal backdrop that expands and contracts--it literally breathes as the song goes along.  It's a genius touch that only a genius could execute.
Although Music from Big Pink is a hard album to cherry pick for favorites, this one is a standout.