Preheat oven to 180oC/160oC fan-forced. Grease three 7cm-deep, 20cm (base) round cake pans or three 4cm-deep, 20cm base round cake pans. Line bases and sides with baking paper.
In a large bowl using an electric mixer, beat butter, vanilla and sugar until light and fluffy. Add eggs, one at a time, beating after each addition. Stir in flour, baking powder and milk until combined.
Divide batter between 6 bowls. Tint 1 portion blue, 1 portion green, 1 portion yellow,1 portion orange, 1 portion purple and the last red. Spread 1 portion into 1 prepared cake pan. Repeat with remaining 2 portions and prepared pans. Smooth surfaces. Bake for 25 minutes, rotating halfway during cooking, or until cooked through. Stand cakes in pans for 5 minutes. Turn out onto wire racks to cool completely. Wash and dry cake pans. Grease and line with baking paper. Repeat with remaining cake mixtures.
To make frosting, place eggwhites, vanilla, sugar and 1/4 cup cold water in a large heatproof bowl. Whisk to combine. Place bowl over a saucepan of simmering water. Using an electric hand mixer, beat eggwhite mixture for 15 minutes or until stiff peaks form.
Remove bowl from heat. Beat until cooled.
Place purple cake on a 30cm round cake board. Spread with 1/2 cup frosting. Top with blue cake. Repeat with remaining frosting and green, yellow, orange and red cakes, using 1/2 cup frosting for each layer. Spread remaining frosting over top and side of cake or decorate as desired with chocolate ganache or royal icing.