With our many trips through the "Joy" that is Chicago traffic, we have tried all different times, many to no avail (I'm NOT kidding, it literally takes TWO HOURS to get through that place!). After the third time of making it in a record breaking 30 minutes, I am happy to report that if you hit it between noon and 1:00 pm, you can, for the most part, make it through relatively unscathed. I can also tell you this, Chicago looks WAY better at 50-65 mph then it does at 10 mph.
So, if you have a trip coming up that takes you through the traffic turtle that is Chicago, do whatever you have to, leave early, take along a portapotty, anything to hit the magic moment, and I guarantee you will have hair left by the end of the road trip (as opposed to pulling it all out during the dreaded drive!) :).