Place the blackberries in a large bowl, and mash a few of them with a large spoon or fork. Stir in peaches and quick-cooking tapioca. Taste to see if the fruit needs any sugar. Set aside, and make the topping.
FOR THE TOPPING
Whisk the brown sugar and the flour together in a large mixing bowl. Add butter, cutting it in with a pastry blender or a fork until the mixture is crumbly. Stir in oats and cinnamon, blending until clumps form.
Place the fruit mixture in a 9x9-inch pan, and top with the crumble. Bake in a 350° F oven about 30-40 minutes, or until filling is bubbling and topping is brown and crisp. Let cool at least 15 minutes before digging in.