This little bit of weirdness was recorded during the White Album sessions, and John indeed lobbied hard for it to be included on that album. When time constraints got the song voted off, he tried to release it as a single by the Plastic Ono Band, his new band (really just his backing band at any given moment), but his plan to put "You Know My Name (Look Up the Number)" on the B-side irritated the other Beatles (since it was, you know, their track) and the whole plan got the kibosh. But that might give you an idea of how enamored John was with this song.
I think what was happening here is that John had been hanging out with Yoko Ono and her artist friends, and my admittedly limited understanding of the avant-garde scene at the time is that there was this value of some kind of a "pure" idea being the perfect art, or something. So whatever you thought of first, was necessarily your best idea, at least if you took this to its logical extension. (I think Paul was guilty of this way of thinking too-- I am not enamored of his habit during his solo career of splicing together little musical doodles, just because there's theoretically some kind of purity in the doodles as doodles, rather than just finishing a few damned songs using his powers of revision. But anyway.) So here in "What's the New Mary Jane" John has written basically one musical line-- the line in the verse-- that he repeats over and over again with different bizarre lyrics, all of which I'll bet he came up with in like two minutes, him being John Lennon and all. He threw in this eerie chorus almost just for the hell of it. Then he and Yoko and George (the only three people on this track) started in on making noise, and whatever happened was just going to be awesome and random and art.
So is it? I mean, I don't know. I like a bit of surrealism and weirdness and noise as much as anyone-- what else is "I Am the Walrus," after all?-- but this kind of thing runs a very high (and very obvious) risk of sounding stupid, pretentious, and lazy all the same time. And that's what John's done here, I think. And what kills me is that if he had just finished the song and figured out how to, you know, orchestrate it and structure it and so forth, it might have been awesome. There are moments in the lyrics that are great and creepy (shades of "Happiness is a Warm Gun" and "Come Together," don't you think?), and if he had edited himself a little there too he could have come up with a whole song full of zingers. But no-- John thought this was great enough as it was to release a single. I know he was trying to break free from the Beatles with new and different sounds and all, but I would argue that this wasn't the best way to do it. (His first solo album, Plastic Ono Band, is FANTASTIC, possibly the greatest thing a solo Beatle ever did-- and it sounds like nothing the Beatles would have ever put out. So, I mean, it's possible to dodge the Beatleness and still make a good song, is my point.)
"What's the New Mary Jane" was bootlegged for years in various forms-- there are a few different versions-- before the studio mastertrack became available on Anthology 3. Over the years the song attracted quite a bit of curiosity, probably just because of its weirdness, and one of the legends that sprouted up about it is that Syd Barrett co-wrote it. I think this is based only on the fact that someone realized he was in Abbey Road studios that same day, but it seems to not be true at all, as no one who was actually there during this little jam session has ever brought it up. Besides, it sounds all John to me, much as I would like to have someone else to blame.
"What's the New Mary Jane," released in the U.K. disc 1 track 22 of Anthology 3, October 28, 1996.