Navigate to any site that you want to remain signed in to. Enter your username and password as usual, and then select the provided checkbox option to stay signed in. It will be labeled with "Remember me," "Stay signed in" or other wording to the effect that selecting it will allow you to stay signed in.
Is there a way to stay signed in to Google?
Go to the “Settings” section. Tap on “You and Google,” then “Sync and Google Services.” Enable the “Allow Chrome sign-in” option.
Why do I have to keep signing into Google?
If Chrome keeps logging you out, it's most probably because of this one setting: Turn off Clear cookies and site data when you quit Chrome. You find this under Settings > Privacy and Security. Please make sure you turn it off.