I'm prompted on every reboot for a password for a stale iCloud account
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 &gt; ~/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.