At which Olympic Games did professional athletes first take part in competition?

This may not ever be possible to answer definitively, as there may have been professional athletes who nevertheless did compete even when professionals were banned. Jim Thorpe won gold medals in 1912, but was later stripped of them because he was found to have played some semi-professional baseball before those games. The medals were later reinstated, both because attitudes towards professionalism had changed, and because the sport in which he achieved his medals was not related to the sport for which he had been paid. During the 20th century the ways of getting around the ban became blatant pretenses, and in 1986 the IOC decided to allow some professional athletes to compete.

The 1988 Seoul Olympics were the first ones to allow professional competitors (with the proviso that they were not yet 23 years old), but only in the sports of tennis (this was the year when Steffi Graf won a Golden Slam, winning all four of the professional Grand Slam tournaments, as well as Olympic gold) and soccer. In the 1988 Calgary Winter Olympics, ice hockey players under the age of 23 were also allowed to compete.

http://www.nytimes.com/1985/03/01/sports/olympics-to-allow-pros-in-3-sports.html

The trial was deemed a success, and the IOC subsequently moved to allow any and all professional athletes to compete.

https://en.wikipedia.org/wiki/Olympic_Games#Amateurism_and_professionalism