I migrated from Exchange 2010 to 2013 over the weekend. I cannot log into the EAC with my domain administrator account I use to log into all my other servers. I also cannot run the clean-mailboxdatabase cmdlet logged in as this user. I had no trouble moving mailboxes from the old server to the new server with this account though.
This account is a member of: Domain Admins, Enterprise Admins, Exchange Full Admin, Exchange Organization Admin, Organization Management, Schema Admins, Server Management.
I can log into the EAC with another admin account that has the same memberships as the Administrator account.
I tried giving the account the role of "Databases" as suggested by others to fix the clean-mailboxdatabase issue but that did not work for me either.
The Administrator mailbox has been moved to the new database on the Exchange 2013 server. The Exchange 2010 has been decommissioned and is turned off.