That is a lot of chess, but there have been other threads on this and well, you're not going to believe the numbers. A few years ago there were several gentlemen from India with over 1,000 current games; the highest had 1600 going at the same time! But is that the highest? No. Last year there was a gentleman from the Phillipines who peaked at something over 3,000. That was just a bit bogus however as periodically he would time out on hundreds of games. The guys from India OTOH seemed to actually be keeping up with their games.
Is there a record for the most current games?
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I came across someone on this site who has finished 211 games, has 541 current games, and has been a member since 28 July 2014.
If you play turn-based games, 7 days per move, that's about 10 moves per hour, 8 hours per day.
Anyway, that seems like a lot of chess to me. I'm impressed.