I just went through all the answers here so far and right now I'm salivating.

Then, as a friend asked me to add an answer too, I thought of picking out some fruits from 'my garden', not mentioned in here yet and which I don't remember seeing in North America.
Yes, it is a good tactic to limit myself with this 'my garden' idea, Raquel.. In Brazil we do have way too many 'exotic' fruits, and I say exotic even for us brazilians. I consider Amazon and North East of Brazil where they have the most exotic and sweet fruits. But I'm from South..
See if you have seen these ones:

Jabuticaba (Myrciaria Cauliflora) - Some call it Jaboticaba a name that comes from Tupian language: 'fruit bud'

It's 100% brazilian and quite exotic as they grow attached to the tree's trunk.
I think this fruit belongs to a childhood time of many and it used to be sold only by kids or adults on the roads, just like they sell tamarind in remote places of Turkey.

But what's news for all of us is what searcher Daniela Brotto Lopes Terci from Laboratory State University of Campinas, São Paulo, just found out:
Our dear Jabuticaba contains huge amounts of anthocyanins. More than grapes or berries. Also its white pulp is rich in iron, phosphorus and niacin.
Yay!

(Web Image)

Araçá (Psidium Araca Raddi)

Araçá tastes almost like Feijoa (Goiaba da Serra as we call it in Brazil, 'Guava of the Mountain'), but if you haven't tried Feijoa it will remind you Guava.
Not explored commercially yet but very rich in vitamins A, B and C plus protein and carbohydrate.

I just took this picture. When they mature they turn yellow. Thing is, it's a baby tree and I have already eaten the ripe ones this morning. They were very sweet :)

Pitanga (Eugenia Uniflora) - from Tupian language pi'tāg meaning 'vermelho'= 'red'

Another fruit not commercially explored yet, they say it is difficult to sell it in nature because they are too delicate, but come on, what would you say of Raspberry then?
Native to Mata Atlântica (The Rain Forest) which once upon a time covered most of coastal Brazil, pitanga is rich in vitamin C and calcium.
Such a beautiful fruit with an unique shape, texture and alive red color. The flavor is sharp and acid but not tart if eaten ripe.

Besides eating it fresh, you may make pitanga juice, pitanga jam, pitanga liquor... Well, in the case of the liquor, some say it's aphrodisiac, so watch it out.

(Web Image)

Jaca (Artocarpus Hererophyllus) - Jackfruit

This fruit makes me smile as in my walks to school I used to pass by this neighbor who had this immense tree with the jacas hanging on.. I remember wondering how could they afford to hold themselves there being so big and heavy. The 'jaca dura' type can reach from 15 to 40 kilos! I'm not sure about other ends but jaca is the biggest fruit in Brazil.
It is not native but brought from Asia in 18th century and now it feels home, specially around the indians in Minas Gerais.
Full of carbohydrates, rich in vitamin A, C, thiamin, niacin, iron, sodium, calcium, fiber, phosphorus and goes on.

There's a curiosity in here:
"(...) In 1984 the searcher Maria Cristina Roque Barreira discovered the 'jacalina', substance made from jaca's seed. But in those days Brazilian legislation did not permit the registration of products made from vegetal. And so 'jacalina' was not patented.
But it is produced and sold by biotech companies practically worldwide. Brazil never received anything for that. The jacalin is a biochemical reagent that identifies whether or not the mother has antibodies in breastmilk to pass to the newborn, which ensures the baby immunity (...)" - TV Cultura site.

Besides eating it fresh which is starchy and fibrous, you may make sweets and even eat the seeds that can be sweet too. In some places I heard they salt the seeds and dry them in the oven, just like salty nuts, making a good snack.

Unfortunately that immense tree from my teen days is gone, a way too sad urban reality in Brazil. But the other day I couldn't help and stop to take a hurried photo from this baby all chopped out to fit in this minuscule forgotten piece of land surrounded by concrete. You can see fruits hanging from inner trunks or the trunk itself. Funny looking, isn't it? (Sorry photo quality)

Pinhão (from Araucaria Angustifolia)

Maybe I'm wrong in adding pinhão (a pine nut) in this list, but many consider pinhão a fruit and the gralha azul (azure jay) and tiriva (tiriba) will clap their wings for me on that.
Almost harvesting time now as Autumn is around (April/May).
Oh they are versatile and for many, delicious. You can use in desserts, just boiled cook, main dishes, tuck them inside your wood oven among the logs for a little.. Last year I got addicted to this: after boiled, I peel and halve them. On a frying pan I add a line of olive oil (very little as pinhões aren't porous) and add the halves with a little salt: Delicious as an appetizer.
And I discovered that the birds are right, it can be eaten even raw but that no one will tell you because it is a walk in the woods thing.

Do you in North America have Green Coconut yes? But not the habit of drinking its chilled water on the streets, at the beach or after a work out or or or.
Gosh I don't understand how the Caribbeans sell it at room temperature. Once I told this seller he would sell it much more if chilled. He looked at me like "Pwef, these tourists..."
Oh well.

PS.: I just noticed jackfruit is in Vladimir and Andrew's list.