Even though trips like this often start at $14,000 per week per couple (bottomless Champagne and caviar included), there is no shortage of takers. Big-name resort brands including Four Seasons and Aman—along with an influx of high-end concepts from billionaires such as Bernard Arnault of LVMH Louis Vuitton Moët Hennessy SE and François Pinault of Groupe Artémis—are meeting the demand with new ships.