Instructions

For the raspberry syrup: In jar, combine ⅔ cup sugar, ⅔ cup water, and raspberries. Cover jar with lid to seal. Shake jar vigorously until sugar dissolves, about 2 minutes.
Let jar sit on counter for 30 minutes to infuse flavor.
Place fine-mesh strainer over bowl. Remove lid from jar and pour syrup mixture through fine-mesh strainer into bowl. Discard solids in strainer. Pour flavored syrup back into jar. (Raspberry syrup can be refrigerated in airtight container for up to 1 month.)
For the frozen raspberry-lime rickeys: Cut 1 lime in half through both ends. Lay lime halves, flat side down, on cutting board, then cut each half crosswise into thin semicircles (see photo, “Step-by-Step: How to Slice Limes,” below).
Add lime slices and 1 cup sugar to large bowl. Use potato masher to mash sugar and lime slices together until sugar is completely wet, about 1 minute. Set aside.
Cut remaining 6 limes in half crosswise. Use citrus juicer to squeeze lime juice into bowl with sugar and lime slices.
Pour 4½ cups cold water into bowl and use rubber spatula to stir mixture until sugar is completely dissolved, about 1 minute.
Set fine-mesh strainer over large pitcher. Carefully pour mixture through strainer into pitcher. Use rubber spatula to stir and press on limes to get out as much juice as possible. Discard lime slices in strainer.
Carefully pour half of lime mixture into 2 ice cube trays. Place trays in freezer and freeze until frozen solid, 2 to 3 hours. Place remaining lime juice mixture in refrigerator.
When limeade cubes are frozen, pop them out of ice cube trays and add to blender jar. Pour remaining lime juice mixture and 6 tablespoons raspberry syrup over top.
Place lid on top of blender and hold firmly in place with folded dish towel. Turn on blender and process until smooth, 30 to 60 seconds. Stop blender. Pour into glasses and serve immediately.