The Field Museum of Natural History has always been my Museum Campus favorite ever since I was a kid. The Field Museum moved to its current location on the Campus in 1921, courtesy of its largest benefactor, Marshall Field (hence the name).
The Field Museum's collection of biological, anthropological, natural, and historical items is one of the largest and finest in the world with over 20 million specimens. The museum also hosts excellent touring temporary exhibits.
For more information, including hours and admission prices, read my full Field Museum profile.