Yes, the adapter is in the center console and is a USB port that you plug the iPod into. No other equipment needed. It has been this way since since I owned my 2008 550.
Can't speak for the '13 M3 or the new Nano. I keep a dedicated Nano (older gen) in the center console of my 2011 M3 and it stays charged through the white iPod cable plugged into the USB port.
What I am not sure of is the lightning/USB cable. I would think the Nano would charge just like mine does but I cannot confirm this...
2011 Space Grey E90 M3, DCT, Comp. Pkg.
1997 Z3 2.8, Artic Silver, Tanin Red interior, MT
1974 2002tii, Inka, MT