Another vote for the Talking Heads Stop Making Sense. It just starts out great and gets even better with each song.
And a very honorable mention to the Who on Live at Leeds and to Cheap Tricks Live at Budokan. Both of which were very influential to me in my youth.
And of course no list would be complete without Neil Young and Crazy Horse performing Live Rust. I saw them perform it live and it was as spectacular as the album and the movie.
At our FCA!35 show, at the Chicago Theater, in between songs, as Frampton and his band were preparing for the next song, someone came walking onto the stage from the side; I initially assumed it was a roadie (and I think that Frampton did, too). Nope, it was a very attractive young woman, who walked up to Peter, and kissed him, before security caught up to her, and whisked her away.
He was stunned for a moment, then laughed. “Does she realize how old I am?”, he asked the crowd. “Though, she was very pretty.”
My answers are going to be hard, and very, very subjective. Most of the time, I’m one of the people who enjoy the more “polished” songs from the studio album. So for me, a live album is usually an inferior format. Not, NOT throwing shade at those who have different preferences, just that mine are based entirely on how a particular live show or album affected me.
My personal favorite is from John Denver’s various Concerts at Red Rocks from the mid-70s. Perhaps because those become the core versions found on later Greatest Hit albums that were a big part of my musical life in the early 80’s car trips with my folks. And I enjoy the various jokes and cracks he makes during several of my favorite songs during the live performances.
Another favorite (taken from the top 50 list link above) is U2’s ‘Under a Blood Red Sky’ (1983), another Red Rock location, but for very different reasons. I didn’t much get into U2 until the late 80s when I was personally more in charge of my musical playlists, rather than those chosen by my parents, but outside of classic protest songs, these were some of the first that spoke to me. And the visuals from the '83 concert between the brooding weather and lyrics were greater than the sum of their parts.
Later yet, one of the few albums where I preferred the Live version to the studio would be “Somewhere in the Stratosphere” by Shinedown (from their 2011 tours). Again, with this particular band and energy, the live just works better and has more energy.
And of course, a honorable mention to any number of Broadway cast recordings of various musicals, which probably shouldn’t qualify as “albums” but I wanted to mention anyway!
I doubt any of my person mentions, mostly being quite niche and less than timeless should or do qualify for “Greatest Live Album of All Time” (well, unless you’re a big John Denver fan like me), but those are the ones that work best for me. Subjective, after all. 
There are far too many to make a definitive pick as the Greatest, but I’ll throw a couple more out that haven’t yet been mentioned:
Live Bullet - Bob Seger and the Silver Bullet Band, 1976
The Last Waltz - The Bank, 1978
An Evening With John Denver is one of my favorites. Thanks for mentioning JD.
Only three live albums ever got repeated use on my turntable:
The Beatles at the Hollywood Bowl - a truly horrible recording, but at age 13 I could crank up the volume, pick up my guitar, close my eyes, and pretend I was John Lennon in front of 50,000 screaming girls.
Cheap Trick at Budakan - actually mostly recorded at a smaller venue in Osaka. A great performance in front of an enthusiastic crowd, the highlight being their cover of Fat’s Domino’s “Ain’t That a Shame”.
The Ramones, It’s Alive - 28 songs in less than an hour, IIRC.
I was going to also add The Last Waltz , but my version was by The Band. I didn’t realize the film version was directed by Martin Scorsese.
Rory Gallagher / Irish Tour ’74
If you haven’t heard of Rory before, after listening to this or seeing the video from the tour, you may end up re-ordering your list of best live guitarists.
Dammit!!
“Cheap Trick At Budokan” is great, as is Rush’s first live album, “All The World’s A Stage.”
Always fun to find a fellow fan, both of the artist and album!
Which is also the answer to “Greatest Live Concert Movie of All Time? What do you think?”
Then I’d add The Last Waltz as well…
Yes, I’m aware that the Dead album and (to a much lesser extent) the Stones album had some studio fixes.
Oh, yeah - Rory was a great live act. Not an awesome lyricist, I don’t think he had the best songs in the world. But he was always entertaining, especially on an acoustic IMO. His early power trio Taste also had a pretty decent live album from their performance at the Isle of Wight.
Looked for Cheap Trick, Peter Frampton and Talking Heads on the list, which is good (although I think Frampton deserves a higher ranking). I also love the Fela Kuti one with Ginger Baker, although it’s not remotely my favorite Kuti material. Keith Jarrett at Koln is also great.
Missing from the list: Laurie Anderson’s United States Live. It was perhaps too niche for this list but it was absolutely groundbreaking stuff.
IMHO, one of the best live albums ever was Leonard Cohen – Live in London at the O2 Arena in 2008, which was recorded and filmed and released as both an album and a video. IIRC at that time Cohen was more or less retired but had to go on the road again when it turned out that his manager had embezzled most of his retirement funds.
Nick Cave and the Bad Seeds : Live Seeds
The Cure : Show
Both 1993 releases.
A couple not mentioned yet that hold a dear place in my heart:
If you listen to each of their live albums, you can see their progression of tempo. Faster, and faster, and faster. The 1 2 3 4 was yelled in record time at the end.
I’ve never been that enamored of live albums but one I have listened to a lot is KISS Alive! . I would never claim this is the greatest and I also know they did a lot of studio work on it (I’m guessing that isn’t unique for “live” albums), but if you like early KISS, this live album is great.
Pretty funny coming from a guy with an avatar of Bob Dylan and a couple of live albums mentioned above that he is part of. I do like them as well.
I have to say that I think that Cheap Trick Silver is one of the best filmed concert performances ever. The album of it is good, but not great.
Now, give me a chance to pick a live album that always brings a smile to my face and I’m going to go for The Mothers - Fillmore East June 1971. Always makes me feel 60 years younger.
I would have lots more love for The Allman Brothers’ album if my drunk roomie didn’t ALWAYS come back late and put “Whipping Post” on at 120 dB. That song has been ruined for me.