Cucumber-Basil Agua Fresca

Slice a few thin slices of cucumber; reserve for garnish. Peel and coarsely chop remaining cucumber. Purée chopped cucumber, lime juice, basil, 1/4 cup sugar, and 2 cups water in a food processor until very smooth. Strain into a pitcher and add 2 cups water, adding more sugar or water if needed, then chill until ready to serve. Add reserved cucumber slices just before serving.