I currently have games-in-progress in 7 games on Steam and in 5 games in my GoG library, all of them being games with story/campaign modes. As far as multi-player goes some of those same games do have an online mode I play occasionally as well. I also play three MMROPGs lately. In just one day I can play short sessions of many of them, or depending on how I feel or if I have enough time to dedicate to just one long session then I'd progress just one or two of my games. I'd say that it just generally depends on my mood, but I usually don't play more than 2 or 3 different games in one day. There's always exceptions though, for example during the past two weeks I've played something like 15+ different games, seven of which have been recently-purchased (Steam sales oblige). I have enough time, however. I'm still in vacation for two weeks.
The one game I've been progressing at least once every day since the past week so far is Tomb Raider (2013). I only play short sessions, sometimes I don't actually progress the story but simply travel back to previously-visited areas to complete the maps and find all the secrets. I think I would have been able to complete it in about three days maybe if I had only played that one. But since I play it maybe 45 mins per day or so then it's taking me a while, but it's ok I'm used to consume my game campaigns slowly to let the fun persist.
I also have other game's campaigns which I started two months ago and only play them maybe once every week, Saints Row 4 (still not completed and it's still my first playthrough) and TW: Shogun 2 (only my second full campaign since I bought it) being two such games I only play at such intervals. It varies a lot in my case, that is unless you bring me a game like the original Mass Effect, or Dragon Age: Origins. In such cases I would play every day, every available hours, until the realization that I forgot to eat and sleep for a week. That did not happen since Skyrim which I've played a good 4 days almost without pause a week after it was released (except eating less than usual and sleeping maybe two nights out of that marathon).