You can refreeze ice cubes and other ice as many times as you want to refreeze them.
Ice cubes are just water that has been frozen to create the ice cubes from water.
The ice cubes can be refrozen several times and will not affect the ice cubes at all.
When I take out some ice cubes from my ice cube tray I fill the ice cube tray back with water in place of the ice cubes and put it right back in the freezer so I always have ice cubes available.
The water turns into ice cubes through freezing where the liquid water cools down and the water molecules also slow down.
As the water molecules slow down due to freezing the water molecules arrange themselves into a rigid and crystalline structure and forms ice.