I faced this error last year when we tried to move to EasyEngine, but unable due to this error, now having again. Message we are getting
Error on Functon *table_name* - Access denied for user 'www-data'@'localhost' (using password: NO)
Things we have tried
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Give all permission to the DB User created by EE
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Create a new user
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Disable the plugin that use that specific table. Then it hangs in the next table
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Tried to connect to our remote db server from EE install, same issue.
 
So, its not a permission issue.
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Basic WordPress install works fine.
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We tried clean cookie, change timezone, update.
 
Site is on Latest WordPress 4.2.3 and we are testing on EE 3.3.3. And its almost basic EE setup with w3tc.