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Any way to turn off the radio?

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Don't know if this works on 2013, but you could try to unselect the radio as a Source. Also, if you can do this, likely another Source will replace it, and you would have to reselect it if you wanted to hear it again.
 
Dial the volume down to 0 or use the voice commander "Mute Media".

A little more complicated - on a USB drive, save a single mp3 audio file that is blank (i.e. no sound). Attach the USB drive to either of the two front console ports or the glovebox port if equipped (newer Model S). Then to turn off audio, select the USB source or use the Voice command "Switch to USB".
 
Sorry, I misread what you wanted. I'm not aware of a way to completely make the audio screen disappear. You can make it very small by swiping down from the top of the audio box (at least on the 2021+ Model S). It's been 8 years since I sold by 2013 S, so I don't remember if there is a similar "shrink" option. Try swiping down at the top edge.
 
What SW your car is at? I think this is the “half screen radio page” that OP is talking about, see the photo. Can’t you minimise the radio window (as you would on your computer) by holding the top part of the window and dragging it to the bottom of the screen? This is how it works on mine, MCU1 03/16 S.
p.s. one click on LH scroll wheel will act as MUTE

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But it reappears the next time you drive it.
Yes. I join you in finding it annoying. On the legacy model S/X you can close it, but it is always there at the start. On the refresh it is even worse because there is no way to close the audio box, ever. I don’t have the spring update yet and am hoping against hope that they have updated the refreshed S/X to be like the 3/Y with a bit more control.