As far as I know, anyone can play as many games as they like. The restrictions come in the form of the learning tools, and being able to join and create tournaments.
Number of Open Games for Free and Paid Members

Yeah one one the main selling points for chess.com is that all chess (live and turn-based) is free and unlimited for everyone including basic members who don't pay anything. There are some restrictions for tournaments however where non premium members are limited to a certain amount.
I think you should visit this page if you want to find out more: http://www.chess.com/membership.html?
I am a also a member of another Internet chess site. The free members are restricted to no more than 6 simulataneous games plus one possible extra game if engaged in a mini tournament (maximum of 7 games).
Sliver level members ($6.99 per month) are restricted to 30 simultaneous games plus up to 4 additional games, one for each mini tournament involved (maximum of 34 games).
Gold level members ($39.95 per year) are restricted to 50 simultaneous games plus up to 8 additional games, one for each mini tournament involved (maximum of 58 games).
Platinum level members [$59.50 per year (my current level) are restricted to 100 simultaneous games plus up to 12 additional games, one for each mini tournament involved (maxinum of 112 games).
Titanium level members ($89.50 per year) are restricted to 200 simultaneous games plus up to 16 additional games, one for each mini tournament involved (maximum of 216 games)
My inquiry is as follows:
How many simultaneous Chess.com games can be played for each classification at this site -- free, Gold, Platinum, and Diamond?
Thank you for any help you can give me with this.
Best, Jim