Some of our fave freebies in Chicago were
The Money Museum (the kids could never get enough of this place!)
Chicago Cultural Center (mostly art exhibits, but sometimes they have events or performances too)
Home Depot's Crafts for Kids (if I'm not mistaken, there's one this Sat. too)
I believe the zoo has enough indoor exhibits to make it worth a trip too, but we never went there. We were closer to the Brookfield Zoo and just got a membership and practically lived there. ;) My trick is to get a membership to one place each year out of my tax return, then we go there like weekly all year. We get out and have a great time every week, and by the end of the year the kids know the place like the back of their hands so we agree on a new attraction. In Chicago, you could cycle like that with just the huge attractions and go 5+ yrs between repeats! =D
All the museums, planetarium, aquarium, etc. in the town have free days too. Some are often, some are only 1-2 a year, but there are usually a lot happening over Christmas break to encourage families to do enriching things with the kids while they're out of school.