I am arrived in Baku! Here goes:
1. Chess player: Garry Kasparov. Maybe the greatest player of all time? He is not ethnic Azerbaijani, but grew up in Baku.
Teimour Radjabov. It is amazing for how long he has gotten away with playing the King’s Indian Defense at the highest levels of chess competition.
Shakhriyar Mamedyarov. Over the last year, he has had the best results of anyone in the chess world, including Carlsen. His forcing style resembles that of Kasparov.
Vugar Gashimov. He was pretty good too, passed away prematurely in 2014.
Cellist and conductor: Mstislav Rostropovich, born in Baku. His Bach Cello Suites are perhaps my favorite of all extant recordings. Here is one (different) YouTube version. As a conductor he was uneven, but capable of spectacular live performances of Shostakovich.
Philosopher: Max Black, also born in Baku. He edited Frege and worked on problems from Leibniz, such as the identity of indiscernibles.
Note that numbers 1, 5, 6, and 7 on this list were Jews who emigrated to America.