If you have kids or like to climb on stuff, the
St. Louis City Museum. Don't be mislead by the word "museum"; perhaps "jungle gym and art installation" would be a bit more accurate. People seem to either love it or hate it.
A few around Vermont, where I live...
The large and fairly well-known
Shelburne Museum (with a large and eclectic collection, tending towards folk art).
The somewhat less well-known
Fairbanks Museum in St Johnsbury (natural history mostly, but also a sort of time capsule or museum museum—you get the feeling it hasn't really changed a whole lot in the past century. They have a cute little planetarium there, too.) Nearby in St. Johnsbury is the public library or
Athenaeum with a fine little art gallery.
The
American Precision Museum in Windsor has a fine collection of historic machine tools. It's been a looong time since I was there; I need to get back sometime.
Circus Smirkus is a vermont-based youth circus that tours around New England each summer with a proper one-ring bit top tent. It's generally well worth seeing.