Brigitta Sinka, an 87 year old Hungarian chessplayer, has broken the record for the most number of simultaneous games played in a career, having just clocked up her 13,600th game. More correctly, she has broken Jose Capablanca's record of 13,545 games, which is believed to be the previous record.
Of course such claims are sometimes difficult to prove, but in her case she has kept a record of the 13,600 games she has played, starting back in 1957. A lot of the games were played at holiday camps for children, but I suspect even then the quality of opponents would be reasonably high, being Hungary and all.
Nonetheless I would not be surprised if there are players who have played more games than this, just without the requisite documentation. One name that springs to mind is Joseph Henry Blackburne, who made his living from simultaneous displays. He had a very long career stretching from 1862 to 1921, and it is estimated he played 100,000 games over this time. Assuming he played simuls for 50 years, and 20 board simuls were typical, he would only need to play 20 simuls a year to clock up 20,000 games. But based on his own collection of best games it seemed he played many more than that annually, so even 20,000 seems to be an underestimate.