I keep an off-site backup at my parents house. Right now it includes a printed copy of my backup codes, so if my house burns down and everything is a total loss here, I at least don't have to start from zero. They live far enough away that my backup offsite backup can get to be a few months out of date but that's usually fine. (If I were to make a major change in something I'd make a special visit)

I don't want to spend a bunch of time when I visit to find that key and add it to all of my new accounts and hope I got everything - I want to make a backup of my current key right before I visit and when I visit, I just put the new backup key in the desk drawer and take the old one home with me.