Gary Barnett has his hands full with the recent $170 million acquisition of the Helmsley Carlton House hotel on Madison Avenue, the looming threat of millions of dollars in forced rent refunds at his luxe Upper West Side renovation project, the Belnord, and a flurry of lawsuits from buyers trying to back out of their contracts at his 80 Riverside Boulevard condominium, the Rushmore. Still, the Extell Development head has benefited from a cautious approach to real estate over the past decade, putting down 20 percent on many of his boom-time acquisitions, while his peers were going in on as little as 5 percent. So despite his recent challenges, it’s not a stretch to say that Barnett is among the most prolific real estate developers in the city, as Crain’s does in a profile of the developer this week. His roster of current projects alone includes the almost-finished 535 West End Avenue condo, the 34-story International Gem Tower going up at 44 West 47th Street, the massive Riverside Center complex planned for the Upper West Side waterfront, to name a few. As for his personal life, a cagey Barnett refused to tell Crain’s his age (53, sources say), his wife’s name (Ayala, according to public records), or how many children he has (more than five, say the sources).