One Saturday night last September, a crowd of celebrities, Silicon Valley titans and Hollywood executives gathered at television producer Mark Burnett’s Malibu mansion for a birthday party celebrating venture capitalist Vivi Nevo.

The richest of them all was Amazon.com Inc. founder Jeff Bezos, whose stake in Amazon is worth about $125 billion. He sat in the middle of the action and appeared to relish the spotlight, said a guest who knows Mr. Bezos.

A...