Patchy

Patchy lives on your desktop. Not in a window on your desktop, actually on it, in front of your wallpaper and behind your cursor, doing his own thing while you work, play games or whatever it is you do. You can click straight through him. Your icons, windows, taskbar, all of it behaves like he isn't there.

He finds his own things to do

Leave him alone and he'll go fishing, water the garden, cook something, paint, build a model kit, drag a telescope out and look at the stars, chase bugs with a net, feed the birds, work out, and occasionally just dance for no reason at all.
He's not good at all of it either. Cooking he actually has to learn.
Early on he burns dinner and then a little drone puts out the fire. Over time all of his skills will improve. 
Whatever he finds or makes, he keeps
Fish go in the aquarium. Insects get pinned in a display case. Flowers and finished models end up on a shelf. It piles up in his corner of the screen over time.  In the companion app you can decide what's actually on display and what goes into storage.
He pays attention to you

He watches your cursor. He notices when you pick him up, and he has opinions about being thrown. He knows what the weather's doing outside too, once you've given him your coordinates (he can't see out of a window, so you have to tell him), and he'll put an umbrella up in the rain.
And if you drag a file onto him and hold it there for a moment, he will take it off you, look at it, then carry it over to a trash bin and chucks it in. It goes to the Recycle Bin, so nothing is ever really gone.
Things turn up

Cats, mainly. Sometimes one, sometimes five at once, and he'll sit down on the floor and pet whichever is closest. A dog shows up now and then wanting to play fetch, so Patchy will throw the ball and the dog brings it back, and the two of them will keep that going for a little while.
Some of the other visitors are considerably bigger or much scarier than he is and much less friendly. You'll go hours without seeing one of those tho.