Maybe not what you're after given Alinea, but I can tell you if you want Chicago pizza, you want Pequod's (char cheese crust!!) or Lou Malnati's. You need to get a hot dog. It doesn't matter where, as long as it's a Vienna brand frankfurter. Accept no substitutes on this. Italian beef, skip Al's. it's gone all franchised and terrible. Portillo's isn't bad and there's a number of locations. And you can get your hot dog there too.
Another cheaper place (I don't generally do white-tablecloth joints) is Big & Little's sandwich shop. They serve foie gras fries, which is a nice way to give yourself that 3rd heart attack and not totally break the bank.
There's also White Castles there (as in Harold and Kumar go to...) but unless it's 2am and in a suitably altered state it's probably not worth tracking one down. (They closed the one on Addison, which made me very sad)