Preheat oven to 350°F. For the Cake, prepare 1 package of the cake mix as directed on package. Divide batter evenly among 3 bowls (about 1 1/3 cups batter in each bowl). Stir raspberry flavor and 10 drops of the red food color into first bowl. Stir lemon extract and green food color into second bowl. Stir orange extract, yellow food color and remaining 10 drops red food color into third bowl. Pour each color of batter into separate greased 8-inch round cake pans.
Bake 15 minutes or until toothpick inserted into center comes out clean. Cool on wire rack. Cut out circles from each cake with 1-inch round cookie cutter.
Prepare second package of the cake mix as directed on package, stirring in vanilla. Divide batter evenly among 3 greased 8-inch round cake pans. Press assorted colored cake circles into batter in each pan. (Cake circles will be visible.)
Bake 15 to 20 minutes or until toothpick inserted into center of white cake comes out clean. (Cake circles will remain visible.) Cool cakes in pans 10 minutes. Remove from pans; cool completely on wire rack.
Meanwhile, for the Frosting, beat butter and vanilla in large bowl with electric mixer on medium speed until light and fluffy. Gradually add confectioners' sugar alternately with milk, beating well after each addition and scraping sides and bottom of bowl frequently. (If frosting is too thick to spread, gradually beat in additional 1 tablespoon milk.) Fill and frost cooled cake with frosting.