Lucian Freud’s final portrait is of a naked man and a dog. It is unfinished but otherwise betrays no sign of the agedness of its creator, who died last July 20, halfway through his 89th year. The scale is big, a square canvas of about five feet by five feet, and the brushwork is as sure and layered as in any painting he had ever done—smooth and free around the man’s shoulders, crusty and impastoed along the arms. The palette is Caucasian-fleshy from afar but remarkably varied and intricate up close: purples and greens in the man’s legs, vivid streaks of yellow in his right hand, rust and blue at the naughty bits.

For the last 57 years of his life, Freud painted standing up rather than sitting down; the physical restrictions of seated painting, he said, had begun getting him “more and more agitated” in the 1950s, so he kicked the chair away. Painting on his feet required extraordinary stamina, given Freud’s self-imposed work schedule: a morning session with one model, an afternoon break, and an evening session with another model, seven days a week, all year round. What’s more, these sessions had a tendency to stretch on: a deliberate worker, Freud took 6, 12, 18 months or longer to complete a painting, marathoning into the night if the mood struck. But he had stamina in spades. Painting was his workout; he took no other exercise, and yet photographs of him working shirtless in 2005, when he was 82, show him to be lean and all sinew, a jockey-size Iggy Pop.

But by June 2011, Freud recognized that his body was finally failing him, and that he had only so many brushstrokes left. The naked man in the portrait was completed, but the dog, a tan-and-white whippet, would never get its hind legs. Freud prioritized its head and face, adding a little dart of terre verte (“green earth”) mixed with umber to depict the tip of the animal’s pricked-up right ear. In early July, Freud was addressing the painting’s foreground: the folds and ripples in the ...