What are some ARs/FMs that a soldier should carry a hard copy of?

A cheap android or windows based tablet with the PDF versions is the way to go. You have everything in your pocket. You would need a good protective case though to bring it to the field.

Being that they are revised so frequently these days, it's hard to say. As a leader, you bounce from reg to reg as you search for information. Thank God we live in an automated age.

I don't know that I'd carry hard copies, but bookmarking apd.army.mil on your smartphone will get you access to all of them. I reference AR 600-20 (Command Policy) often. And if writing is part of your job, AR 25-50. A well formatted memo is a sign of a detail oriented, squared away Soldier.