A year after chef Grant Achatz opened Alinea here in 2005, Gourmet magazine named the restaurant the best in the country, and the prestigious Mobil guide gave it its highest rating of five stars. The young chef’s exotic, lavishly presented creations — a mango duck dish is served on a deflating pillow that releases lavender-scented air — have connoisseurs lining up to book meals that cost an average of $240 a person. But last month, doctors gave Mr. Achatz, 33 years old, devastating news. A cancerous tumor was growing inside his tongue. The disease was so advanced that three doctors told him the only way to cure it was to cut out part of his tongue, leaving one of the world’s most celebrated chefs to ponder life without the ability to taste.