Heat oven to 325°F. Generously grease bottoms and sides of each square cake pan using solid shortening. Sprinkle with flour to coat pans; tap to remove excess flour.
To make batter for 6- and 8-inch pans, in large bowl, beat 2 boxes cake mix, 2 1/2 cups water, 1/2 cup oil and 6 egg whites with electric mixer on low speed 30 seconds, then on medium speed 3 minutes, scraping bowl occasionally. Spread 2 cups batter in 6-inch pan and 4 1/2 cups batter in 8-inch pan. Cover remaining batter with plastic wrap; refrigerate.
Place both pans about 3 inches apart on middle oven rack in oven. Bake 6-inch cake 35 to 45 minutes and 8-inch cake 40 to 50 minutes. Cake is done when toothpick inserted in center comes out clean. Cool in pans on cooling racks 15 minutes. Remove cakes from pans; place top sides up on cooling racks. Cool completely, about 1 hour.
Each tier is made up of 2 layers of cake. To make another 6- and 8-inch layer, wash and dry both pans. Prepare pans again as directed in Step 1. In large bowl, beat 1 box cake mix, 1 1/4 cups water, 1/4 cup oil and 3 egg whites on low speed 30 seconds, then on medium speed 3 minutes, scraping bowl occasionally. Add batter to batter set aside in refrigerator; stir to combine. Fill pans as directed in Step 2. Bake and cool as directed in Step 3.
To make batter for 12-inch pan, in large bowl, beat 2 boxes cake mix, 2 1/2 cups water, 1/2 cup oil and 6 egg whites on low speed 30 seconds, then on medium speed 3 minutes, scraping bowl occasionally. Spread about 9 cups batter evenly in pan.
Bake 12-inch cake 50 to 60 minutes. Cake is done when toothpick inserted in center comes out clean. Cool in pan on cooling rack 25 minutes. Remove cake from pan; place top side up on cooling rack. Cool completely, about 1 hour.
Wash and dry 12-inch cake pan. Prepare pan again as directed in Step 1. For second layer, make batter, fill pan, bake and cool as directed in Steps 5 and 6.
To make frosting, in large bowl, stir together 8 cups powdered sugar and 1 tablespoon meringue powder. With electric mixer on low speed, beat in 1 cup shortening, 1/2 cup milk, 2 teaspoons vanilla and 1 teaspoon almond extract; beat on medium speed until smooth. Add additional milk, 1 tablespoon at a time, to make frosting spreadable consistency. Transfer frosting to another large bowl; cover with plastic wrap. Make frosting as directed 3 more times.
To assemble cake, with serrated knife, cut off domed top crust of each cake layer to form a flat surface. Carefully brush off crumbs from cakes.
Place about 1 tablespoon frosting on serving tray where first cake layer will be placed to keep cake from sliding on tray. Place 1 (12-inch) layer, cut side up, on tray. Place large round decorating tip in decorating bag; fill with frosting. Pipe around top inside edge of cake layer, forming a thick frosting border about 3/4-inch high. Spread 1 cup raspberry filling over top of cake inside frosting border. Top with second 12-inch cake layer, cut side down.
For 6- and 8-inch layers, make cardboard square to place cake tiers on by placing bottom of cake pans on cardboard; trace around pan edge. Cut out with scissors. For 6-inch tier, place small amount of frosting on center of cardboard square to hold cake in place. Place 6-inch cake layer, cut side up, on cardboard. Pipe a frosting border around top inside edge of cake. Spread 1/4 cup raspberry filling over top of cake inside border. Top with remaining 6-inch cake layer, cut side down. Repeat for 8-inch tier, using 8-inch cardboard square, both 8-inch cake layers and 1/2 cup raspberry filling.
To frost 6- and 8-inch tier, stir frosting. Using large spatula, carefully spread thin layer of frosting over sides of cakes, then over tops. Use spatula to push frosting, never allowing spatula to touch cake. (The purpose of this technique is to seal the crumbs to make frosting the cake easier.) Apply second thicker layer of frosting to sides and tops of cakes, spreading until smooth.
For a stacked cake where tiers are placed directly on top of one another, each tier will need to be supported by several dowel rods. To mark where to place 8-inch tier on top of 12-inch tier, center 8-inch cake pan, turned upside down, on top of 12-inch tier and gently press to make imprint; remove pan. Insert 1 dowel rod into center of 12-inch tier, straight down to tray. Mark rod with exact height of tier; remove rod. Cut 7 more rods to marked length. Insert rods into 12-inch cake, spacing evenly 1 1/2 inches apart within imprinted outline on 12-inch tier. Place 8-inch tier on top of 12-inch tier.
Mark 8-inch tier with imprint of 6-inch cake pan; remove pan. Cut and insert 8 dowel rods for 8-inch tier as directed in Step 13. Top 8-inch tier with 6-inch tier.
Use remaining frosting to decorate cake as desired. Add royal icing flowers.