The first song that George Harrison wrote for The Beatles was “Don’t Bother Me.” Initially, when George joined the band, he wasn’t a songwriter. He left that duty to John Lennon and Paul McCartney. Since they seemed capable of writing The Beatles’ hits, George didn’t even think about writing songs.
However, George was still curious to see if he could. The result was 1963’s “Don’t Bother Me,” but George was its harshest critic.
“To get it straight, if I hadn’t been with John and Paul I probably wouldn’t have thought about writing a song, at least not until much later,” George explained. “They were writing all these songs, many of which I thought were great. Some were just average, but, obviously, a high percentage were quality material. I thought to myself, ‘If they can do it, I’m going to have a go.'”
So, George set to work writing his first song, although not in the best of circumstances.
Source: cheatsheet.com