Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 10-inch fluted tube pan (such as Bundt(R)).
Combine cake mix, water, eggs, pudding mix, and almond extract in a bowl. Blend on low with an electric mixer until moistened. Increase speed to medium and blend about 2 minutes more. Pour into the prepared pan.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, 40 to 45 minutes. Cool in the pan for 25 minutes; transfer cake to cooling rack.
Combine ricotta cheese, 1 cup confectioners' sugar, white sugar, and vanilla extract in a bowl. Whip cannoli cream until smooth using an electric mixer.
Beat heavy cream in a chilled glass or metal bowl with an electric mixer until foamy. Add 1/4 cup confectioners' sugar and 1 teaspoon vanilla extract at a time to achieve desired sweetness, continuing to beat until cream is stiff.
Transfer cooled cake to a cutting board and cut into small, bite-size pieces. Arrange 1 layer of cake pieces to cover the bottom of a trifle bowl. Top with a layer of chocolate chips and the cannoli cream. Continue with layers until bowl is nearly full; top with the whipped cream and chocolate chips.