There's a green grocer on Devon a few doors from Patel brothers (sorry, I can't recall the name, not enough coffee in me yet, but the one on the corner with the green awning) - they have fresh lychees in season, as well as several varieties of mangos. They are also a good source for fresh figs (green & black). Prices are very reasonable.
I've not noticed fresh jackfruit, but that's because I don't buy it, they may well carry this (they certainly carry frozen & tinned). I'm pretty certain I've seen longan & durian there too, plus they carry green coconut to drink. I believe you won't find mangosteen in Chicago - IIRC its only recently become available in the US & only in NY & LA.