i have 400 and only play like 10 minutes a day
so if you have 4000 (chess.com doesn't allow you to start games if you have more than 10 games to play or more than 5 pending requests and these two combined make it insanely hard to do that)
you would only have to play liek 2 hours a day
not that bad
but i don't really try which is why my rating is so much lower than my blitz ratin for example
59 at the moment. I think my high was in the 70's
He has currently 1792 games running.
Wow, time really does fly. I'm surprised I posted this almost 2 years ago.
I just realized this was replied to a couple months ago.
It's funny in my original post I wanted "like 1000 games". I found a simple way a few months ago to add rapidly to your daily games: you just join daily tournaments.
So now I've found myself with 274 games going on and I am completely overwhelmed. Before I found about these tournaments my timeouts were a solid and consistent 0%. Now they are 3.33%. There's a guy I'm playing in several games (probably 30) named panamacitychess, and initially he had 1700+ games going on with a 1900+ rating, and a very high win to loss ratio. I thought surely he was cheating, but I've since realized his rating has dropped to about 700. Not that this player is a bad player at all, but after being overwhelmed with 274 games I can see how 1700+ games are nearly impossible to maintain thus explaining the harsh rating drop due to timeouts.
So at this point I no longer desire any more daily games. I have wasted (well I can't call it wasted) 2 years of chess playing to basically giant simuls. I'm going to let these games dwindle down to nothing and start playing 30 to 45 minute rapid games again. I do work a full time job and I thought initially this would be the perfect way to be motivated to play many games of chess with 14 days to move. Now that I can't keep up, I've lost motivation to play and many of my games with 2 weeks to move have reached minutes to hours to move. And though it puts me on vacation automatically, this doesn't feel satisfying at all. I would like to keep up. But it's easier said than done. Also, I feel like these daily games have increased my rate of blunders.
I see sleeper61, impressive to keep up with 1200 plus games.
Atom680, do you have a link? 4,000 seems absolutely impossible.
6 years fly by 
Alright it looks like we have a winner here...
https://www.chess.com/member/andrewllll
2500 games with 0% timeout!?!?
Thats huge by all standards , I have 86 games open at present.