The most obvious place was the fortress, a few hundred metres away on the nextdoor island. But as I was scanning the list of local places, I saw Ytterby, on the island of Resarö.
That's two islands away, and required us to take a bus to Engan, change there, and get off at a junction slightly in the middle of nowhere.
(Oh, these islands are quite close together - the bits of sea separating them are sometimes only a few yards wide, and have simple bridges or causeways connecting them. With no tides, and storm waves either, you often don't even notice you're crossing some sea, except for seeing water on both sides.)
We walked down into Ytterby and Ytterbystrand. They're delightful places to have houses - views out over the water and marinas, well spaced houses, all looking extremely prosperous in the summer sunshine.
Oh, so what was so special about Ytterby?
Do you know of any other place in the world that had four different chemical elements named after it? Yes, this small village is the source of the names Yttrium, Ytterbium, Erbium and Terbium.