Which country in the world doesn’t have a native airline? ?

Vatican, San Marino, Monaco, Andorra, Lichtenstein.  

It should be noted that there was a planned Spanish airline called Air Andorra but it never started operations.  Andorra Air was an Andorran airline, but it too never flew.  In any event, there are no airports in Andorra, the nearest airport with commercial service is Andorra–La Seu d'Urgell Airport is actually located in Spain, 12 km south of the border.  

Actually, the US does not have a flag carrier in the sense of national sponsorship.  Technically all US airlines are considered flag carriers but again, the US meaning is different than most countries where one airline is state sponsored or heavily subsidized.  You can find out more info on wikipedia's article on Flag Carriers.  
https://en.wikipedia.org/wiki/Flag_carrier

I'm quite certain the list is extensive.  And just because 'American' is in the name it doesn't mean much.  Are you asking about airlines which are backed by the government somehow or just have an airline which is owned by a company in that country?