If you mean the Tinker to Evers to Chance infield with Harry Steinfeldt playing third, that would be Frank Chance.
Frank Chance - .296
Johnny Evers - .270
Harry Steinfeldt - .267
Joe Tinker - .262
The 2009 Chicago Cubs had a team batting average of .255
in 2007 Jim Thome led the Chicago White Sox in batting with a .275 average.
hornsby
Jermaine Dye had a .274 batting average in 2005 with the Chicago White Sox
Derrek Lee of the Chicago Cubs won the league batting title with a .335 average.
The 2005 National League batting championship was won by Derrek Lee of the Chicago Cubs, who posted a .335batting average.
Riggs Stephenson and Bill Madlock are tied for the Cubs career batting average leader with .336.
Derrek Lee of the Chicago Cubs won with a .335 batting average.
Joe Jackson (Shoeless Joe) with a .340 batting average for the White Sox between 1915-1920.
Hugh Duffy hit .440 in 1894 for the Chicago white stockings.
Ted Williams had .420 under current batting average calculation rules in 1941, but officially he hit .406; the actual record belongs to Tip O'Neill, who hit .492 in 1887. If you discount 19th century players, the record is .424 by Rogers Hornsby.
Paul Konerko batted .283 in 2005 for the Chicago White Sox.