There are a number of exchanges that trade vanilla Call/Put American/European options on various underlyings (equities, indices, futures). There have been some trading in digital options on certain underlyings in US, but I don't see any indication that they are still traded. The ICE exchange lists some Asian APOs (Average Price Options), while both CME and The ICE list some CSOs (Calendar Spread Options) which I would also consider exotic for the purposes of this question.
What other exchange-traded exotic options do you know? Please, specify the name of the exchange and the type of the exotic options that are offered there.