Norman Hsu was desperate for invitations to glitzy Democratic Party galas in California and private political dinners in New York. But once he got in, Mr. Hsu, a 56-year-old apparel executive, seemed awkward and out of place, almost astonished to be posing for pictures with former President Bill Clinton, Senator Hillary Rodham Clinton and other big-name Democrats. He gave generously, showering money on a wide array of national, state and local politicians. But he stood out in the symbiotic world of campaign finance because he appeared to want nothing in return other than a few powerful friends, according to Democratic...