Song you want played at your funeral?

Actually, I always heard and read that it was John's composition. He's also the one who sang it. Paul wrote the melody.
Here's a section on it from Wikipedia:

Composition

According to Lennon, the song's origins can be traced to when the English journalist Kenneth Allsop made a remark that Lennon should write songs about his childhood. Afterwards, Lennon wrote a song in the form of a long poem reminiscing on his childhood years. The original version of the lyrics was based on a bus route he used to take in Liverpool, naming various sites seen along the way, including Penny Lane and Strawberry Field. Those original lyrics are on display at The British Library.
Original lyrics to John Lennon's "In My Life"

However, Lennon found it to be "ridiculous", calling it "the most boring sort of 'What I Did On My Holidays Bus Trip' song"; he reworked the words, replacing the specific memories with a generalized meditation on his past. "Very few lines" of the original version remained in the finished song. According to Lennon's friend and biographer Peter Shotton, the lines "Some [friends] are dead and some are living/In my life I've loved them all" referred to Stuart Sutcliffe (who died in 1962) and to Shotton.

In a 1980 interview, Lennon referred to this song as his "first real major piece of work" because it was the first time he penned personal lyrics about his own life.

Regarding authorship of the melody, Lennon's and McCartney's recollections differ. Referring to McCartney, Lennon said "his contribution melodically was the harmony and the middle-eight itself." McCartney claimed he set Lennon's lyrics to music from beginning to end, taking inspiration for the melody from songs by Smokey Robinson & the Miracles. "I liked 'In My Life'. Those were words that John wrote, and I wrote the tune to it. That was a great one."