Google has announced that, in the name of safety, devices using the older operating system will no longer be able to sign in to some apps. So it’s advising that, if your device uses Android 2.3.7 or older, you should update to Android 3.0 or higher if you’re able.
On and after September 27, you'll likely receive a user name/password error when you attempt to log in or make new accounts for some Google apps. Google also states that you'll receive the error if you perform a factory reset and try to log back in, change your password, create an account, or remove and re-add your account.
It's unclear if you can delay the error via never signing out, but even if that's possible most apps and services require you to log back in eventually.