Did The Beatles write down any of their lyrics or music?

in the studio yes.

and i know this is about the music, but since you brought up massive influence; brian epstein had at least as much to do with the beatles success, particularly in the early days, than anyone ever involved with the band. he brought them from the cavern club to the ed sullivan show.

I just finished this book (thanks for the recommendation), which includes a few specific stories about music and lyrics that The Beatles wrote down. Paul was organized and “always” carried a notebook around with him, in which he methodically wrote down lyrics and chord changes. When John’s portion of “A Day in the Life” was recorded, it lacked a middle section; Paul went through his notebook and discovered a bit of an unfinished song written some time before that he felt might fit, and John liked his suggestion. In contrast to organized Paul, John was always searching for scraps of paper that he’d hurriedly scribbled ideas on. As mentioned, John nearly always used a lyric sheet when recording, which was often taped to a music stand in front of his mic. In planning the fade-out ending of “Good Morning, Good Morning,” John had written out a list of animal sounds that he wanted to hear in order at the end of the song (his list was in fact followed in the final version). And during the Revolver sessions, instead of practicing songs ahead of time, John, Paul or George would come in to the studio with “scraps of paper” with lyrics or chord sequences on them, and within a day or two, the completed song would be recorded.