I used to have a mulberry tree before we had to cut it down.
Similar in texture to raspberries, stain a bit less than cherries, and have their own nice taste. It's fun to shake the tree to get them to fall down, but you need a big cloth to catch them. Make sure to take out any beetles and ants if you shake though, they can get mixed in. They can grow in a lot of places without much hassle too.