score:14
It was William Paterson of New Jersey, who was only the third person to vacate a seat for any reason in the United States Senate when he resigned on November 13, 1790 to become the second govenor of New Jersey. He was preceded by William Grayson of Virginia who died while holding office on March 12, 1790 and John Walker (also of Virginia) who was appointed to the post when Grayson died and held it until a successor was elected on November 9, 1790. Moses Robinson of Vermont was the 14th senator to resign.
Since there doesn't seem to be a list of U.S. Senators that have resigned anywhere (and I was bored), I compiled one here using Wikipedia - 312 total Senators with 332 resignations at the time of this post by my count. Note that this includes 3 people who were elected to the Senate but resigned prior to being seated (denoted by an asterisk).
William Paterson NJ November 13, 1790
William Samuel Johnson CT March 3, 1791
Richard Lee VA October 8, 1792
Charles Carroll MD November 30, 1792
George Read DE September 18, 1793
James Monroe VA March 27, 1794
John Taylor of Caroline VA May 11, 1794
James Jackson GA November 16, 1795
Oliver Ellsworth CT March 8, 1796
Rufus King NY May 23, 1796
Caleb Strong MA June 1, 1796
George Cabot MA June 9, 1796
Jonathan Trumbull, Jr. CT June 10, 1796
Moses Robinson VT October 15, 1796
Richard Potts MD October 24, 1796
Pierce Butler SC October 25, 1796
Frederick Frelinghuysen NJ November 12, 1796
William Cocke TN September 26, 1797
William Bradford RI October ??, 1797
Isaac Tichenor VT October 17, 1797
John Henry MD December 10, 1797
Philip Schuyler NY January 3, 1798
John Vining DE January 19, 1798
Andrew Jackson TN April 1, 1798
John Sloss Hobart NY April 16, 1798
John Hunter SC November 26, 1798
John Rutherfurd NJ December 5, 1798
Joseph Anderson TN March 3, 1799
Samuel Dexter MA May 30, 1800
John Laurance NY August 1, 1800
Benjamin Goodhue MA November 8, 1800
James Lloyd MD December 1, 1800
James Schureman NJ February 16, 1801
Henry Latimer DE February 28, 1801
Ray Greene RI March 5, 1801
Charles Pinckney SC June 6, 1801
Samuel Livermore NH June 12, 1801
Elijah Paine VT September 1, 1801
John Armstrong, Jr. NY February 5, 1802
James Sheafe NH June 14, 1802
Dwight Foster MA March 2, 1803
DeWitt Clinton NY November 4, 1803
Theodorus Bailey NY January 16, 1804
John Armstrong, Jr. NY February 23, 1804
Abraham Venable VA June 7, 1804
John Armstrong, Jr. NY June 30, 1804
William Wells DE November 6, 1804
William Giles VA December 3, 1804
Andrew Moore VA December 3, 1804
John Breckinridge KY August 7, 1805
Robert Wright MD November 12, 1806
John Adair KY November 18, 1806
David Stone NC February 17, 1807
James Fenner RI September ??, 1807
Israel Smith VT October 1, 1807
John Smith OH April 25, 1808
John Quincy Adams MA June 8, 1808
Samuel Maclay PA January 4, 1809
Aaron Kitchell NJ March 3, 1809
Daniel Smith TN March 31, 1809
John Milledge GA November 14, 1809
Buckner Thruston KY December 18, 1809
Nahum Parker NH June 1, 1810
James Hillhouse CT June 10, 1810
Return Meigs, Jr. OH December 8, 1810
Thomas Sumter SC December 16, 1810
Jenkin Whiteside TN October 8, 1811
Christopher Champlin RI October 12, 1811
Jean Noel Destréhan LA October 1, 1812
James Bayard DE March 3, 1813
Dudley Chase VT March 3, 1813
William Crawford GA March 23, 1813
James Lloyd MA May 1, 1813
Chauncey Goodrich CT May 13, 1813
George Campbell TN February 11, 1814
Michael Leib PA February 14, 1814
George Bibb KY August 23, 1814
Thomas Worthington OH December 1, 1814
Jesse Bledsoe KY December 24, 1814
David Stone NC December 24, 1814
William Giles VA March 3, 1815
Francis Locke* NC December 5, 1815
William Barry KY May 1, 1816
Christopher Gore MA May 30, 1816
John Taylor SC November ??, 1816
William Wyatt Bibb GA November 9, 1816
James Turner NC November 21, 1816
Robert Goodloe Harper MD December 6, 1816
Jeremiah Mason NH June 16, 1817
James Fisk VT January 8, 1818
George Campbell TN April 20, 1818
Eli Ashmun MA May 10, 1818
George Troup GA September 23, 1818
John Forsyth GA February 17, 1819
John Crittenden KY March 3, 1819
John Eppes VA December 4, 1819
Prentis Mellen MA May 15, 1820
Walter Leake MS May 15, 1820
William Logan KY May 28, 1820
James Wilson NJ January 8, 1821
Freeman Walker GA August 6, 1821
Harrison Gray Otis MA May 30, 1822
John Williams Walker AL December 12, 1822
James Pleasants VA December 15, 1822
Caesar Rodney DE January 29, 1823
Samuel Southard NJ March 3, 1823
James Brown LA December 10, 1823
Ninian Edwards IL March 3, 1824
Henry Johnson LA May 27, 1824
James Barbour VA March 7, 1825
David Holmes MS September 25, 1825
Andrew Jackson TN October 14, 1825
James DeWolf RI October 31, 1825
Edward Lloyd MD January 14, 1826
James Lloyd MA May 23, 1826
William Henry Harrison OH May 20, 1828
Albion Parris ME August 26, 1828
Thomas Cobb GA November 7, 1828
Nathaniel Macon NC December 14, 1828
Ephraim Bateman NJ January 12, 1829
Mahlon Dickerson NJ January 30, 1829
John Berrien GA March 9, 1829
John Branch NC March 9, 1829
John Eaton TN March 9, 1829
Louis McLane DE April 16, 1829
Edward Livingston LA May 24, 1831
Issac Barnard PA December 6, 1831
Powhatan Ellis MS July 16, 1832
Littleton Tazewell VA July 16, 1832
Robert Hayne SC December 13, 1832
William Marcy NY January 1, 1833
George Troup GA November 8, 1833
William Rives VA February 22, 1834
John Forsyth GA June 27, 1834
Ezekiel Chambers MD December 20, 1834
Peleg Sprague ME January 1, 1835
Charles Gayarré LA January ??, 1836
John Tyler* VA February 29, 1836
Ether Shepley ME March 3, 1836
Willie Mangum NC March 19, 1836
Isaac Hill NH May 30, 1836
Arnold Naudain DE June 16, 1836
Benjamin Leigh VA July 4, 1836
John Clayton DE December 29, 1836
Alexander Porter LA January 5, 1837
Richard Parker VA March 13, 1837
John McKinley AL April 22, 1837
John Pendleton King GA November 1, 1837
John Black MS January 22, 1838
Felix Grundy TN July 4, 1838
James Trotter MS July 10, 1838
Ephraim Foster TN March 3, 1839
Richard Bayard DE September 19, 1839
Hugh Lawson White TN January 13, 1840
Robert Strange NC November 16, 1840
Bedford Brown NC November 16, 1840
John Davis MA January 5, 1841
Daniel Webster MA February 22, 1841
Clement Comer Clay AL November 15, 1841
Franklin Pierce NH February 28, 1842
Alexander Mouton LA March 1, 1842
Henry Clay KY March 31, 1842
Samuel Prentiss VT April 11, 1842
Samuel Southard NJ June 26, 1842
Reuel Williams ME February 15, 1843
John Calhoun SC March 3, 1843
William Sprague RI January 17, 1844
William King AL April 15, 1844
Nathaniel Tallmadge NY June 17, 1844
Silas Wright, Jr. NY November 26, 1844
John Berrien GA March 1, 1845
Daniel Elliot Huger SC March 3, 1845
Levi Woodbury NH September 20, 1845
William Haywood, Jr. NC July 25, 1846
Walter Colquitt GA February 4, 1848
Ambrose Sevier AR March 15, 1848
Lewis Cass MI May 29, 1848
John Crittenden KY June 12, 1848
Arthur Bagby AL June 16, 1848
John Clayton DE February 2, 1849
Reverdy Johnson MD March 7, 1849
Daniel Webster MA July 22, 1850
Jefferson Davis MS September 23, 1851
Robert Rhett SC May 7, 1852
John Berrien GA May 28, 1852
William King AL December 20, 1852
Robert Stockton NJ January 10, 1853
Solon Borland AR April 11, 1853
Pierre Soulé LA April 11, 1853
Edward Everett MA June 1, 1854
Augustus Dodge IA February 22, 1855
Hannibal Hamlin ME January 7, 1857
Asa Biggs NC May 5, 1858
Hannibal Hamlin ME January 17, 1861
Jefferson Davis MS January 21, 1861
John Slidell LA February 4, 1861
Thomas Bragg NC March 6, 1861
Salmon Chase OH March 6, 1861
Thomas Clingman NC March 11, 1861
Andrew Johnson TN March 4, 1862
James Simmons RI August 15, 1862
Waitman Willey VA March 3, 1863
James Bayard, Jr. DE January 29, 1864
William Fessenden ME July 1, 1864
James Harlan IA May 15, 1865
Daniel Clark NH July 27, 1866
James Guthrie KY February 7, 1868
Reverdy Johnson MD July 10, 1868
James Grimes IA December 6, 1869
Charles Drake MO December 19, 1870
William Kellogg LA November 1, 1872
Henry Wilson MA March 3, 1873
Alexander Caldwell KS March 24, 1873
Eugene Casserly CA November 29, 1873
Adelbert Ames MS January 4, 1874
Lot Morrill ME July 7, 1876
John Sherman OH March 8, 1877
Isaac Christiancy MI February 10, 1879
John Brown Gordon GA May 26, 1880
James Blaine ME March 5, 1881
Samuel Kirkwood IA March 7, 1881
William Windom MN March 7, 1881
Thomas Platt NY May 16, 1881
Roscoe Conkling NY May 16, 1881
Henry Teller CO April 17, 1882
Augustus Garland AR March 6, 1885
Thomas Bayard DE March 6, 1885
Howell Jackson TN April 14, 1886
Jonathan Chace RI April 9, 1889
John Carlisle KY February 4, 1893
Edward White LA March 12, 1894
John Sherman OH March 4, 1897
William Clark MT May 15, 1900
Charles Fairbanks IN March 3, 1905
Joseph Burton KS June 4, 1906
John Colt Spooner WI April 30, 1907
Philander Knox PA March 4, 1909
Fountain Thompson ND January 31, 1910
Joseph Terrell GA July 14, 1911
Joseph Weldon Bailey TX January 3, 1913
Warren Harding OH January 13, 1921
John Nugent ID January 14, 1921
Albert Fall NM March 4, 1921
Josiah Wolcott DE July 2, 1921
William Kenyon IA February 24, 1922
Truman Newberry MI November 18, 1922
Frank Smith IL December 7, 1926
T. Coleman du Pont* DE December 8, 1928
Charles Curtis KS March 3, 1929
Walter Evans Edge NJ November 21, 1929
Fred Sackett KY January 9, 1930
Cordell Hull TN March 3, 1933
Claude Swanson VA March 3, 1933
Sam Bratton NM June 24, 1933
Hugo Black AL August 19, 1937
Dixie Bibb Graves AL January 10, 1938
A. Harry Moore NJ January 17, 1938
Frederick Steiwer OR January 31, 1938
William Gibbs McAdoo CA November 8, 1938
Matthew Neely WV January 12, 1941
John Miller AR March 31, 1941
James Byrnes SC July 8, 1941
Henry Cabot Lodge, Jr. MA February 3, 1944
Homer Bone WA November 13, 1944
Sinclair Weeks MA December 19, 1944
Monrad Wallgren WA January 9, 1945
Harry Truman MO January 17, 1945
Harold Burton OH September 30, 1945
Happy Chandler KY November 1, 1945
Warren Austin VT August 2, 1946
Hugh Mitchell WA December 25, 1946
Vera Bushfield SD December 26, 1948
Alben Barkley KY January 19, 1949
Robert Wagner NY June 28, 1949
J. Howard McGrath RI August 23, 1949
Raymond Baldwin CT December 16, 1949
Sheridan Downey CA November 30, 1950
Ralph Brewster ME December 31, 1952
Richard Nixon CA January 1, 1953
Charles Daniel SC December 23, 1954
Hazel Abel NE December 31, 1954
Strom Thurmond SC April 4, 1956
Price Daniel TX January 14, 1957
John F. Kennedy MA December 22, 1960
Lyndon Johnson TX January 3, 1961
Hubert Humphrey MN December 29, 1964
Pierre Salinger CA December 31, 1964
Harry Byrd VA November 10, 1965
John Williams DE December 31, 1970
George Murphy CA January 2, 1971
William Saxbe OH January 3, 1974
Alan Bible NV December 17, 1974
Wallace Bennett UT December 20, 1974
Howard Metzenbaum OH December 23, 1974
Marlow Cook KY December 27, 1974
J. William Fulbright AR December 31, 1974
Edward Gurney FL December 31, 1974
Sam Ervin NC December 31, 1974
Stuart Symington MO December 27, 1976
John Pastore RI December 28, 1976
Walter Mondale MN December 30, 1976
Paul Hatfield MT December 12, 1978
James Pearson KS December 23, 1978
Wendell Anderson MN December 29, 1978
Clifford Hansen WY December 31, 1978
Edmund Muskie ME May 7, 1980
John Durkin NH December 29, 1980
Richard Stone FL December 31, 1980
Harrison Williams NJ March 11, 1982
Paul Tsongas MA January 2, 1985
David Karnes NE December 8, 1988
Dan Quayle IN January 3, 1989
Gordon Humphrey NH December 4, 1990
Pete Wilson CA January 7, 1991
Kent Conrad ND December 14, 1992
Al Gore TN January 2, 1993
Lloyd Bentson TX January 20, 1993
David Boren OK December 15, 1994
Bob Packwood OR October 1, 1995
Bob Dole KS June 11, 1996
Frank Murkowski AK December 2, 2002
Jon Corzine NJ January 17, 2006
Trent Lott MS December 18, 2007
Barack Obama IL November 16, 2008
Joe Biden DE January 15, 2009
Ken Salazar CO January 20, 2009
Hillary Clinton NY January 21, 2009
Mel Martínez FL September 9, 2009
John Ensign NV March 3, 2011
John Kerry MA February 1, 2013
Max Baucus MT February 6, 2014
Tom Coburn OK January 3, 2015