On July 12 1968, fiddler Jack Fallon would enter the EMI Studios as he was hired to fiddle during Ringo's new song at the time: "Don't Pass Me By". The fiddle part was added to the last track of the tape so the bass part (which was erased) had to be re-done by Paul. Some of the piano (with leslie speaker added) had to be re-done as well and this time frame is probably where that piano intro/tinkling came in for the commercial take of the song. At the same session, a mono mix of the song was completed (remix 4) and given to Ringo to take home. This is one of the songs included on a compilation of very early "White Album" mono mixes that was put on cassette and given as a gift from Ringo to actor Peter Sellers. Eventually, the early mixes were copied and released as a bootleg LP/CD entitled "The Peter Sellers Tape" as well as the CD "Unsurpassed Masters, Volume 4". The early mono mix includes some stray bass notes played during the intro to the song as well as the repeat of the first verse which was not included on the commercial version.
There are also differences between the mono and stereo mix of "Don't Pass Me By" on the commercial Apple LP/CD "The Beatles"(White Album). The mono version is sped up by a semitone compared to the stereo mix. Also, there is a different fiddle improvisation between both mixes. This is probably due to two difference takes of the fiddle overdub.
Lastly, there is an orchestral introduction to the version of "Don't Pass Me By" on the Apple 3LP/2CD "Anthology 3" version. The orchestral piece is entitled "A Beginning". This piece remains a bit of a mystery. The piece was recorded on July 22 1968; ten days after the final recording of Ringo's song. The piece also makes an appearance in the "Yellow Submarine" film near the beginning of the Liverpool scenes. Only George Martin knows for sure why this piece exists. To my ears, I don't think the original intent was to have the song lopped onto "Don't Pass Me By" as there is no musical connection; it may just have been experimental. Or as suggested by others, it may have been George Martin's attempt to have this introduction piece included on the "Yellow Submarine" soundtrack LP.