I keep manual backups of the TOTP codes for all accounts, using a QR reader to decode the QR code they give out and saving the result in an encrypted file.
A variant of the Google Authenticator source code is available at https://github.com/google/google-authenticator but is not what's on the iOS/Android app stores.
Open source, lightweight, extremely small app size
It adds intelligent features over Google auth, including encryption of the database, and lets you break down into folders - which makes it infinitely more usable than Authenticator once you've accumulated enough sites.
I irritate very fast from apps that use 4 clicks when 2 would have done, or add other needless steps or hurdles. After around a year of flawless use I can't think of a single aspect that irritates. That's exceptionally rare for me!
It doesn't have ads, even in the free version. I gladly paid, even though I didn't need the few extra features the paid version adds. I just wanted to support very good software. There's a macOS version too, which I haven't tried.
Authy, on the other hand, was uninstalled before adding even one 2FA as it wanted phone and email before allowing you to do anything at all.
[1] https://itunes.apple.com/us/app/otp-auth-two-factor-authenti...