It depends on how much time you spend per move and how much time you want to spend making chess moves in your day, plus what the time control is.
If you play 24 hour games for example, 10 games at 10 minutes per move would take up more than an hour and a half.
Basically, the more time you want to spend per move (which is a good thing) the less games you should play. Playing with 3 or more days per move (time control) will also increase the amount of games you can play simultaneously.
I've gotten up to almost 40 but that's way too many. I'm playing 17 now and don't want to go over 20 max. BTW I'm retired so I probably have more time than most members with jobs and/or family and/or a life away from the chessboard and PC
Playing 10 or less is a good idea so you can put more effort into fewer games. Avoid the temptation to start too many games!
I like to keep it under 20-30 so I can play somewhat decent games. But sometimes I forget about matches I've registered for and I'll go over. For example I forgot I registered for the 20th Chess.com Quick Knockouts tournament and all of a sudden I have 10 additional games.
15 is probably my limit. I could at the outside handle 20. Anything above that and my quality of play would drop a lot.
I envy all of you. 3 or 4 is about my max and to be truthful I'm really only 100% comfortable with 2. I tried 5 or 6 a year ago and went into burnout mode. Oh well, never claimed to be the brightest bulb on the tree....
Are you familiar with up-arrow notation?
Upper limit:200-250
lower limit: 100-140
Right now I have 8 games, but I think I can handle 10. What is the maximum number of correspondence games that you can play simultaneously?