This Friday through Sunday is the annual "sales tax holiday" for the state of Missouri for school supplies. Clothing, school supplies, computers, software, and computer peripherals are all included in the holiday. Full information on what counts for the holiday and price limits is here.
Cities NOT participating can be found here and counties NOT participating can be found here. Unfortunately there are quite a few cities and counties opting out, and several in the St Louis area, so be sure to check before you shop!
From what I can tell, Illinois doesn't have a sales tax holiday although a bill to enact one was passed in the Senate and was being reviewed in the House earlier this year. So hop across the river and do your shopping in the west! :)