Calendar constantly requiring Google authentication on 17.10

I was having the same problem and resolve it by deleting the configuration files of Evolution. Do this:

rm -rf ./.config/goa-1.0/accounts.conf
rm -rf ./.config/evolution
rm -rf ./.local/share/evolution

I don't have that problem anymore since I did that. I hope this could help you.

Sorry to hear you had to revert @Mena. I had the same upgrade path and ran into this as well. Hopefully the below will be helpful for someone else who finds this.

What I found is that the Online Accounts feature isn't fully compatible with Google's TFA. The following is a workaround, to be entered when prompted for a password:
Enter an application specific password for Google in one of the recurring password prompts. Since setting an application specific password, it hasn't bugged me again, even after several reboots and my calendar was syncing before reboot. However for it to sync again- you have to manually bring up the calendar on each reboot, and when it prompts again for password - re-enter the app specific password.

To get the app specific password - go to Google's App password page and generate one to use with Ubuntu's Online Accts (and save it in your favorite password manager for use again later): https://security.google.com/settings/security/apppasswords

For more reading on Google TFA and application compatibility, here's Google's help page on the matter: https://support.google.com/accounts/answer/185833?hl=en

Screencap from me setting an app specific password in Google: