I always worry that adding 2FA could make your machine inaccessible if anything happens to google-authenticator in this case. Maybe it's a little bit of paranoia but I don't like the idea of giving control over my ability to log into my server.
It'd be perfectly reasonable if libpam-google-authenticator relied on Google's infrastructure, but despite the infelicitous name, it does not; it just implements the server side of TOTP. The authentication flow is identical to any other correct TOTP implementation, and you can use any compatible client; no integration with Google services or infrastructure is required. (In fact, I don't think it's even possible.)
Speaking of TOTP without Google, if you use iOS and find the Google Authenticator app unsatisfactory, try Authy. It's good stuff, and well worth a few bucks.
While I'm not in the market for an OTP client, this is the network tools app I've been looking for since I bought my first iPod Touch. If you have remote beer-buying support, point me to it.
I moved most of my 2FA/TOTP stuff to 1Password. Works great on the Apple watch, and it's one less app (I tried Authy, but I have this thing against using too many apps)
I've been thinking about starting to use a password manager (as I should've done long ago), and 1Password looks pretty good. It's pricey enough that I'd like to hear some pros and cons before I pull the trigger, though. How's it worked out for you? Where are the pain points?
FWIW, the desktop app has a free trial, and on top of that, the developer offers a 30-day money-back guarantee if you buy direct from their store. (Not sure about the mobile app, haven't used that.)
I use the desktop app every day and am generally happy with it. It has many thoughtful touches, like automatically clearing the clipboard a minute or so after you copy a password into it, so that you don't later paste your password into some other text field by accident.
google-authenticator is a fancy name for an algorithm described in RFC 6238. It does not talk to servers or require any network access. Google is not involved in any way; they just made the algorithm popular and branded it.
I didn't read the question as "what if Google discontinues authenticator" but more as "what if I use Google authenticator on my phone and it gets stolen and i can't log in any more"