I'm often in longer telephone conferences and like to play relatively uncomplicated, mindless games like 2048 or threes. Both of these are getting pretty boring these days, so I'm looking for new games.
I don't know if you'd call them completly mindless, as they're puzzle games of a sort, but both Mini Metro and Mini Motorways are great relaxing time killers to me.
Not really complicated, just making connections. A fair bit of downtime just everyone make their trips, relaxing music, simplistic visuals, easy to pick up and play.
Infinite Loop (∞ Loop), Flow, Desert Golfing, Simon Tatham's Puzzle Pack, and various card games are some of the games I play just to kill time or keep myself stimulated while listening to something.
Flow Free, and its spinoff games Bridges, Hex, Warps, and Fit.
The puzzles are kind of mindless and not that difficult, but I've really enjoyed it. You do have to pay to get rid of ads, but after doing so the user experience is excellent.
And then what? Stare on a blank screen for 4h a day?
It's my job to be in conference calls where I only listen for 75% of the time and there's nothing else for me to do except listening. And I find it easier to listen when playing around.
Doodling on paper is just as mindless, but that's probably okay for you, because it's "real" right?