AFAIK Hungary is trying something like that, I am not sure about the specifics but I think that for every child you have the government lowers your income tax and after your third child you don't pay any income tax. There are some other measures too, all of them were implemented relatively recently, let's say 2015 although I am not sure.

As for the results, I guess it's still early to tell: https://data.worldbank.org/indicator/SP.DYN.TFRT.IN?location...