Although not a clear indication of maximum GOP performance, here is the Presidential breakdown for 2012 in each district:
District 1: Bobby Rush (D) - 79% D/21% R (last voted GOP for Congress in 1932)
District 2: Robin Kelly (D) - 81% D/19% R (last voted GOP for Congress in 1956)
District 3: Dan Lipinski (D) - 56% D/43% R (last voted GOP for Congress in 1904*-ancestral 4th)
District 4: Luis Gutierrez (D) - 81% D/17% R (created in 1992 by conglomeration of Chicago districts)
District 5: Mike Quigley (D) - 66% D/32% R (last voted GOP for Congress in 1994)
District 6: Peter Roskam (R) - 53% R/45% D
District 7: Danny K. Davis (D) - 87% D/12% R (last voted GOP for Congress in 1924)
District 8: Tammy Duckworth (D) - 57% D/41% R (last voted GOP for Congress in 2010)
District 9: Jan Schakowsky (D) - 65% D/33% R (last voted GOP for Congress in 1946)
District 10: Bob Dold (RINO) - 58% D/41% R
District 11: Bill Foster (D) - 58% D/41% R (last voted GOP for Congress in 2010)
District 12: Mike Bost (R) - 49.7% D/48.2% R
District 13: Rodney Davis (R) - 48.6% D/48.95% R
District 14: Randy Hultgren (R) - 44% D/54% R
District 15: John Shimkus (R) - 34% D/64% R
District 16: Adam Kinzinger (R) - 45% D/53% R
District 17: Cheri Bustos (D) - 58% D/41% R (last voted GOP for Congress in 2010)
District 18: Darin LaHood (R) - 37% D/61% R
District lines change every ten years. I notice that the 3rd and 4th have elected Republicans more recently than 1904, though I don’t know what the borders of the district were at the time.