Willie Keeler won the National League batting title in 1897 at .424 without hitting a home run. Rod Carew won the American League batting title in 1972 at .318 without hitting a home run.
For a season, that would be Pete Browning, who batted .457 in 1887 in the American Association (yes, that was a major league) but finished behind Tip O'Neill who batted .485 .
For a career, that would be the hapless Joe Jackson, who hit .3558 for a career but never won a batting title.
forever
Chicago White Sox, 88 years without winning a World Series. Before 2005 they did not win a World Series since 1917.
dallas cowboys
Winning the war without laying a Finger on his opponent.
UConn Huskies women
President Gerald R Ford
Yes, although it's very risky and not as likely to win without winning the majority of their vote.
buffalo went 4 times in a row with out winning
The last 16 is when they usually fail
Julius Boros
ford
Greg Norman
The Lombardi Trophy is present to the winner of the Super Bowl. It is impossible to win the Lombardi trophy without winning the Super Bowl.
John Q. Adams, Rutherford B. Hayes, Benjamin Harrison, and George W. Bush were elected without winning the popular vote.
forever
The House of Representatives
Your GPA is your "grade point average" aka, all of your grades averaged together. On a regular schedule (one without advanced placement classes and without serious extra credit) a 4.0 is the highest. a 3.3 is still good.