I am a native Chicago guy, with an intense passion for Barbeque, and have found this discussion interesting but feel that some key points have been left out.
Just about all of the places mentioned so far have either been in the downtown area or the Northern suburbs. Neither location is conducive to finding good barbeque which the best in Chicago is found either on the South Side in dicey neighborhoods, or in the SW suburbs which are a huge hotbed in places like Hickory Hills.
In the NW suburbs, I do also like the Great Texas Barbeque in Algonquin as several people have mentioned, but their Ribs and sandwiches are rather unremarkable. Their best item are the smokehouse chicken wings which are spicy, meaty, and have a nice smoked flavor.
In Hickory Hills, the two best places are Baby Joe’s BBQ Co. which has won several awards and is very good. And my personal favorite is "The Pit" which is outstanding wood pit BBQ. The Ribs have a sweet spicy sauce and are nice and tender. The pulled pork is addictive. They also have a very nice wood grilled Greek Chicken.
I have tried most of the chains in the area (Carsons, Robinsons, Houstons) and agree with the assessment that they are uninspired. As a sidenote for those traveling the length of the state of Illinois, that at the Southern end in places like Murphysboro and the like, there is a tremendous volume of quality BBQ. Several teams that have won Memphis in May are from that area, and have long since retired from competative events but have opened up their own restaurants.