My original post was about journeys which are equally or more direct and quick by bus, but which are cheaper.I think the Arriva Day Rover ticket is now £6, but even so it would be quite an endurance test undertaking a day return by Arriva Buses from Holyhead to Chester, I suspect by the time you got to Chester you would more or less have to start return journey!
You would have to change buses four times in each direction, and with Arriva Buses Wales complex timetables, difficult to navigate website and reliability issues, I wouldn’t risk it!