Max died on Monday, a statement from his son on Wednesday said. No cause of death was given.
Max began his career in graphic design in the 1960s, and his swirly, colourful, instantly recognisable designs epitomised the era's "flower power" art. But thanks to his prolific output and fun, accessible style, Max's work remained well known and popular well into the 21st century.
His portrait subjects ranged from US presidents to Taylor Swift.
In his first flush of success in the late 1960s and early 1970s, his mural-like tableaus of flowers, the cosmos and cartoonish figures — all in vivid colours like turquoise, orange and neon green — seemed to be everywhere. Dorm room walls were plastered with his posters. Household furnishings like clocks and bedspreads sported his designs.
"We will remember his extraordinary creativity, his warmth, his curiosity, and the way he saw beauty and possibility everywhere," his son Adam Max said in the statement announcing his death.
"His art became part of American culture, but the man behind that art — the father we knew and loved — is the person we will miss most."
Max was born Peter Max Finkelstein in 1937 to a Jewish family in Berlin. They fled Nazi Germany for Shanghai when he was a baby and later spent time in Tibet, Israel and Paris before settling in New York City when Max was 16. By then, he said in an interview, he was already "a huge fan of American culture": comics, movies and "all the jazz music".
Max studied in New York at the Art Students League, the School of Visual Arts and Pratt Institute. In 1961, he opened a graphic design studio with friends. His bold, unique style was an immediate hit with ad agencies, publications, the corporate world and in pop culture.
Max weathered various controversies in his personal and professional life. He pleaded guilty to tax fraud in 1997 after the IRS alleged he concealed more than $US1 million ($A1.4 million) in income from his art.
He was originally sentenced to two months in prison but ultimately was allowed instead to serve the time in a work-release program, pay the back taxes and a $US30,000 fine, and perform 800 hours of community service teaching art in Harlem schools.
In 2015, his family life became tabloid fodder in a back-and-forth between his second wife Mary and Adam, his son from a previous marriage. The son and a guardian alleged Mary Max had depleted his finances and browbeat him. She in turn claimed that they had kept him from her against his will and stolen paintings that belonged to her. Accounts of court appearances described him as frail.
Max's first marriage to Elizabeth Nance ended in divorce. In addition to his second wife, Mary Max, and his son Adam Cosmo, he is survived by a daughter, Libra Astro.
