For reasons that will become apparent, the most appropriate answer to this question essentially boils down to semantics. Britannica, for example, defines athletics as ‘a variety of competitions in running, walking, jumping and throwing events’, whereas Merriam Webster defines an athlete as a person who is trained or skilled in exercises, sports, or games requiring physical strength, agility, or stamina’.
Applying the first definition, strictly, the athlete who has made the most Olympic appearances is Spaniard Jesús Ángel García Bragado. Born in Madrid, on October 17, 1969, he made his Olympic debut in the 50-kilometre walk at the Olympic Stadium in Barcelona on August 7, 1992; he finished tenth in a time of 3:58.43, exactly eight and a half minutes behind gold medallist Andrey Perlov. Thereafter, García Bragado competed in the same event of the next seven Olympic Games, culminating with the delayed 2020 Summer Olympics in Tokyo. On August 6, 2021, at Odori Park in Sapporo, capital of the Japanese island of Hokkaido, he finished thirty-fifth in the 50-kilometre walk, at the age of 51.
Applying the second definition, though, it can be argued that pistol shooters, whose event does, after all, require physical strength and stamina, should be described as ‘athletes’. In that case, the athlete who has made the most Olympic appearances is, in fact, Georgian woman Nino Salukvadze, who made her Olympic debut, representing the Soviet Union, in the 25-metre pistol in Seoul, South Korea on September 18, 1988. She won the gold medal in that event and the silver medal in the 10-metre air pistol event two days later, having set a new world record in the qualifying round. Like García Bragado, she went on to compete at the next eight Olympic Games, for a total of nine appearances altogether.