Created in 1902 by Edward VII, the Order of Merit is the personal gift of the monarch.
There can only be 24 living members at any one time, making it a highly exclusive collection of figures from the arts, sciences, culture and military.
Current members include David Hockney, Sir David Attenborough, Betty Boothroyd, the former Commons Speaker, and Sir Tim Berners-Lee, the pioneering computer scientist.
The new appointments, the first since 2015, were awarded by the King during a service at the Chapel Royal.
The monarch then hosted a pre-lunch drinks reception in the Picture Gallery, where he complimented Hockney on his yellow footwear, worn with matching glasses and a flamboyant suit.
The artist, who used a wheelchair, insisted on standing when he met the King.
Hockney, 85, is preparing to launch an immersive experience which will give visitors the opportunity to engage with audio and visual elements to follow a journey through his art.
The show will launch early next year at Lightroom, a new four-storey venue in King’s Cross, London.
The King turned to renowned architect Lord Foster and said: “I can’t wait to see what he does next.”