James Madison Museum
Orange, Virginia
Not a lot of "Madisonia" has survived from James Madison -- President #4 -- to stock this museum; First Lady Dolley Madison had to sell most of it to pay off James's debts. This museum compensates by filling out its exhibit space with relics from Zachary Taylor -- President #14 -- who was James Madison's second cousin.
There are a few Madison items, including a smaller version of the head of the big James Madison statue at the Library of Congress; some books from his library; one of Dolley's shawls and an example of her knitting; and Madison's special chair to make siting more comfortable with his rheumatism (It's displayed with a nearly identical Zachary Taylor chair). Madison lived to be 85.
Also here is a cardboard silhouette of the diminutive James Madison -- America's shortest Chief Executive -- a feel-good display that shows most tourists that they are at least tall enough to be President.