Heat oven to 350 degrees F. Grease three 9-inch round cake pans. Dust with flour.

Cake: In bowl, beat butter until creamy. Gradually beat in sugar and vanilla; beat until light and fluffy, 3 minutes. Add eggs, one at a time. Beat well after each.

In a small bowl, mix flour and baking powder. On low speed, beat flour mixture into butter mixture, alternating with milk in 3 additions, beginning and ending with flour, and beating well after each addition. Divide batter among prepared cake pans.

Bake cakes in 350 degrees F oven for 25 minutes or until toothpick inserted in centers comes out clean. Let cakes cool in pans on wire racks.

No-Cook Frosting: In medium-size bowl, mix powdered whites and 1/4 cup warm water; let stand 5 minutes to soften; lightly whisk to dissolve. Add cream of tartar. Beat on medium-high speed until soft peaks form. Gradually beat in sugar, 2 tablespoons at a time, until blended. Beat in salt. Reduce speed to medium; beat in one third of marshmallow creme, then 1 tablespoon warm water. Beat in remaining marshmallow creme, then remaining warm water. Beat in coconut extract; continue beating until frosting is good spreading consistency, adding 1 to 2 tablespoons more water if needed.

Turn cakes out of pans. Place one layer on serving plate (slide thin strips of waxed paper underneath layer to keep plate clean). Spread 3/4 cup frosting over top of cake layer; sprinkle with 2 tablespoons coconut. Stack second layer on top; frost top, sprinkle with coconut. Place third layer on top. Frost sides of cake and top. Sprinkle remaining coconut over sides and top. Remove paper strips. Refrigerate at least 1 hour to firm up frosting slightly.