Garrett Birkhoff (1911-96) (son of George David Birkhoff, 1884-1944, and co-author of the well-known text in O.D.E.'s commonly referred to as "Birkhoff and Rota", among other things). No PhD but was a member of Harvard's Society of Fellows (1933-1936). (This is what I was told but see also the Birkhoff entry at the mathematical geneology page .) Chandler Davis wrote a short obituary for Garrett Birkhoff, as did Philip David.
PhD Students of Garrett Birkhoff: Chandler Davis (see also Chandler Davis) (1950), Pesi Masani (1946) (wrote a biography of Norbert Wiener, obit), Thomas Mullikan, Peter Olver, ...
PhD students of Chandler Davis: John Benedetto (1964) now at UMCP, Zdislav Kovarik of McMaster U., Man-Duen Choi of U. of Toronto, W.S. Tang of the U. of Singapore, ...
PhD students of John Benedetto: George Benke (1971), Wan-chen Hsieh (1977), Fulvio Ricci (1977), Ward Evans (1980), David Joyner (1983), Jean-Pierre Gabardo (1987), David Walnut (1989), Christopher Heil (1990), Rodney Kerby (1990), George Yang (1990), William Heller (1991), Joesph Lakey (1991), Erica Bernstein (1992), Anthony Teolis (1993), Shidong Li (1993), Sandra Saliani (1993), Georg Zimmermann (1994), Melissa Lynn Harrison (1998), Hui-Chuan Wu (1998), Manuel T. Leon (1999), Goetz E. Pfander (1999), and six current students.