He says that when he asked for permission to take her portrait, Kelly replied: "You can have a private audience with me only if you have had a private photographic audience with the Queen of England."
Fortunately he had and the photo session went ahead, but her haughtiness continued, says Davis.
During the audience he asked if he could take a picture of her playing the piano.
She responded sulkily that she didn't play the instrument, yet when she finally agreed Davis recalls that she "began to play like a concert pianist".
In his book, Davis also remembers an occasion when an angry Duke of Edinburgh took offence at his presence at a gala dinner thrown by the Shah of Iran in 1971.
The Duke was heard to call to the Shah: "What's this fellow Davis doing here?"
It was only when the Iranian ruler - deposed in the revolution of 1979 - replied "he's working for me" that the photographer was allowed to stay.
Just as well, as it was at that gala dinner that Davis took the picture of a young Princess Anne, gazing into the distance, which won him Best Royal Picture of the Year. He gave it the caption "A sovereign for your thoughts".
Davis also describes how, when photographing Elizabeth Taylor at the Imperial Hotel in Vienna, the actress began editing his work, selecting which pictures she liked.
"She more or less said, 'One for you and one for me.' With that we left Vienna with half of the material approved," said Davis. "I was not very happy with the situation but had no option but to accept."
Davis, who has spent more than 70 years working as a photographer, says one subject stood out amid all the foot stamping and prima donna behaviour of some of his subjects - the great Italian cinema star Sophia Loren.
He photographed her with her husband, Carlo Ponti, whose marriage to Loren was not recognised by the Italian state since divorce was forbidden in the country at the time.
Davis photographed the couple at their home in Paris in 1975 and when he had finished, Loren kissed him on both cheeks.
"I left with the feeling that the lovely Sophia is not only a great star," he said. "But she is also a lovely human being."
The Grace Kelly fairytale
• American actress Grace Kelly became Princess of Monaco after she married Prince Rainier in 1956.
• She starred in a series of big Hollywood movies, including Rear Window, Dial M for Murder and The Country Girl, which earned her an Academy Award for best actress.
• Kelly and Rainier had three children. One of them, Stephanie, was in a car with her mother in 1982 when the princess had a stroke and lost control of the vehicle. She died a day after the accident.