Most of the early responses to this thread are wrong which is unfortunate as it gives people the wrong impression of Google Play music (esp. for a 2014 thread)...
1) I have found Google Play music to be one of the least restricted music sites both in number of devices and number of downloads.
2) Pinning the music on your mobile device is not really the same as downloading it. "Pinning it" does download true, vague named, mp3 files of your "owned" music which are not encrypted but are kind of hard to find. They can be copied and played without root..but the intent is not for that purpose...you should...see 3
3) download your purchased music through a PC music manager app or now even more conveniently through a chrome app, so Chromebooks can now manage your music at google play...
4) Also you can upload previously owned music to google up to 20,000 files and google will scan and match with a better quality 320kbps copy, so I'm told. Any music purchased at google play does not count towards 20,000.
5) Google play all access is kind of pricey at $9.99, but recent promo 90 day demo for Chromecast owners. It plays radio type mixes but even better allows searching for entire albums and even allows "pinning" them for offline listening. Newer albums I've found don't always have all songs available, probably to encourage purchase.