St Michael's Mount
At the time of my visit (2003), St Michaelʼs Mount Gardens were only open on weekdays in April and May, and on Thursdays and Fridays from June to October. Check with the National Trustʼs site for up to date information.
Do, however, try to get to the island on a day that they are open, if at all possible. Not only are the gardens magnificent, terraces of sub-tropical plants hanging precariously to the side of the island, it also gives one a chance to see the house from the southern side.