I counted about 100672 games in one week (based on a three month average). Quite a nice and round number. It’s almost exactly 10 games per minute.
I think estimating this is quite difficult though… If you pick random games, then about 10~20% of them are played by bots, which have a huge contribution to the whole stack. You’re far more likely to encounter a very active player than a rather inactive one. I took an average of about 10 players, which gave me 110 games per player.
Looking at random players to see if they played the last week or not is also not viable, since you’re far more likely to pick a player who has left long ago, than a player who is active now.
Finally counting the number of players in the last X games and extrapolating that to a week is not a good estimate, since many people play multiple games in rapid succession.
One way to do it, is to take a sample right now of people who are playing games at the moment and look back at a later time to see how much of these people are still there.
Here’s an example of Matt Parker trying to estimate the number of statisticians at a conference: