The University of Chicago is the winner, with 79 Nobel Prizes won by faculty, former faculty, and alumni. As of October 7, 2008 that number is now 81.
The above statement may seem to be the answer to a different question than the one that is asked. That is, many if not most people might think that the questioner really meant "Which university has the most Nobel winners on its faculty"--alumni etc who won Nobels wouldn't count in answering that question.
While I don't myself know the answer to what I believe is the real question, I do know the answer is not Chicago. The University of Chicago has 7 current faculty members who have won Nobels, not 81....see http://www.uchicago.edu/about/accolades/nobel/. Six of these are economists--only one Nobel laureate outside of economics. Berkeley also has 7, although there are 13 other Berkeley faculty who won the prize but have since died. See http://berkeley.edu/news/features/2000/nobel/uc_nobels.html. Harvard has had 43 faculty winners of the Nobel, http://www.news.harvard.edu/guide/faculty/fac7.html, but this also includes those who are deceased or not now on its faculty.
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The Bronx High School of Science graduated seven Nobel Prize-winning scientists -- the highest number of all secondary education institutions -- and six Pulitzer Prize-winning authors.