One new feature of iOS 7 is an activation lock - and this means that you can't sign out or erase the device without entering the proper password for the account to which your device has been locked.
Towards the bottom of the article, it has a link to use My Apple ID web page to check and reset your password if you have forgotten it.
Presumably, if you can't reset the password, you might need to reach out to Apple directly and see if there is an administrative process such as mailing them proof of purchase or correctly answering questions about past billing events to have them reset the account password.
The application icon suggests the password is being requested by Messages.app.
Messages.app
Launch Messages.app and check the accounts listed in the preferences do not include or mention your stale @gmail.com account.
Internet Accounts
Look for legacy accounts in your Internet Accounts within System Preferences.app:
Preferences
Check your preferences for mentions of the old address using the Terminal.app command:
grep -r old@gmail\.com ~/Library/Preferences/*
If this matches a file, modify the following command to examine the file's contents:
defaults read ~/Library/Preferences/MobileMeAccounts.plist > ~/Desktop/output.txt
In the example above, a new file called output.txt will appear on your desktop. This file can be opened with TextEdit.app.
This file may hint as to which application or tool has your old account listed.
If you find a matching preference, consider deleting or renaming it. The preferences will be lost but they should never contain critical information.
Remember to log out or restart after altering the preferences file.
Best Answer
Looks like this answers my question:
https://support.apple.com/en-gb/ios
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