WASHINGTON—Maurice R. “Hank” Greenberg, who built American International Group Inc. into a world-wide financial powerhouse before its controversial government bailout, has donated $10 million to the super PAC backing Republican presidential candidate Jeb Bush, making him one of the largest contributors in the 2016 race.

Mr. Greenberg joins the ranks of a handful of supersize donors. A group of super PACs backing Texas Sen. Ted Cruz has collected some of the largest checks of the election from donors including the Wilks family...