Mulberries are awesome; they're tasty and they're an excellent source of dietary iron, too.

They have two things going against them, though: as fruits they're pretty fragile, even more so than other berries; and when they're flowering, they're highly allergenic for a lot of people. Lots of cities actually ban growing mulberry trees within city limits because of the allergy problem.

Of stuff that grows right in my neighborhood in the Bay Area, California, I'd point out passionfruit and prickly-pears as somewhat unusual fruit.

Passionfruit vines like to grow on fences; they make trippy-looking flowers that mature into lemon-sized fruits full of tasty gooey arils around their seeds.

Prickly-pears are Opuntia cactus, which seem to do oddly well here in even rough and windy coastal areas. The same species can also be harvested for the young cactus pads, which are nopales in Spanish; skin 'em and fry 'em up and put 'em in vegetarian tacos.