1. Peggy Notebaert Nature Museum
Peggy Notebaert is a museum based on science with a focus on the lives of animals and plants. It has special events which include the releasing of butterflies every day at their haven. It is free Thursdays for Illinois residents.
(Courtesy of Benny Mazur at Flickr)
2. Field Museum of Natural History
This museum has many exhibits which include learning about many species of animals and ancient cultures. There are frequent free days and discounts for Illinois residents.
The Art Institute of Chicago is free to residents in Illinois from 5 pm to 8 pm every Thursday. Visitors here can experience all the various cultures (Greek, Roman, Egyptian, Asian and many more) through the collection of artifacts the Institute offers.
4. Museum of Science and Industry
This museum offers a lot on all types of science as well as math by using technology to engaged visitors. The Museum of Science and Industry offers 52 free days throughout the year.
Like the Field Museum, the aquarium has special free days out of the year for Illinois residents. There are plenty of experiences here to see as you walk around to look at the exhibits. Experts on many species of wildlife give presentations on these subjects.
(Courtesy of Dave Pape at Flickr)
Located along Lake Shore Drive, the Adler Planetarium is a great location to check out the solar system with their exhibits and shows. They frequently offer exclusive ways to experience the Adler during the night on certain days.
A more comprehensive list is available at TimeOut for those that are interested.