Is there a way to get a physical disc copy for free of a game I bought digitally?

I'm not sure if EA would be willing to get you the discs; I've never heard of them doing that before....However, you don't need to be connected to Origin to play the games.  I have all my games as digital downloads from Origin, and once I downloaded them onto my computer, I've yet to open the Origin client again.  You only need to be connected to it when you initially download it onto your computer.  After that, you can open your game via the shortcut just as normal.  Are you sure they're downloaded onto your computer?