Far too many... 4+
I've tried to limit myself, and I' happy to say I have 'too many games' nowdays...

One MMO: FFXIV
One singleplayer game.
And as few competitive games as possible: Blazblue, DoA5
Thought atm I've been meaning to get back into Heroes of the Storm and have a crapload of shorter games in my backlog, oh man.

Considering I also go for readable/watchable media at night...yeah I'm kind of juggling a lot of hobbies.