Anyone privileged enough to have flown private knows the appeal. Instead of arriving hours before your flight and enduring long lines, bag-check fees, and a security screening that may or may not involve a full pat-down while your fellow passengers watch, you can pull up to the plane a few minutes before takeoff. Someone parks your car. Someone else takes your bag. Yet another person pours you a drink. You climb a few stairs, settle in, and ascend into the skies as smoothly as a bird takes flight.

But even in this rarified space, there’s a hierarchy, and in their bid to win over the .01 percent, private-aviation companies are learning that it’s not just about getting passengers to 35,000 feet with as few bumps as possible. With private-jet departures hitting a record 3.9 million last year, according to industry tracker WingX, they’re offering increasingly opulent—and occasionally outrageous—perks before, during, and after the flight to lure commercial fliers and poach clients from competitors. Tickets to the Masters? Sure. An in-flight Alice in Wonderland–themed birthday party for your child? Not a problem. Vitamin-infused water for your dog? Only the best for man’s best friend!