Barack Obama’s three closest friends — Valerie Jarrett, Martin Nesbitt and Dr. Eric Whitaker — sat down in the study of Mr. Nesbitt’s house in Chicago for one of their increasingly frequent heart-to-hearts. ... In the presidential campaign, the Obamas had a “no new friends” rule, surrounding themselves with a coterie of familiar faces. Even if the Obamas lift that rule in Washington, newcomers are unlikely to replicate the intensity of this group’s ties, formed over more than a decade ... They vacationed together: in December, with the Obamas in Hawaii; in August, with Ms. Jarrett on Martha’s Vineyard. Mr....