#jQuery 191

i=0;function f(p){$.get('//api.stackexchange.com/2.2/questions/featured?site=stackoverflow&page='+p,function(d){for(x in d.items)i+=d.items[x].bounty_amount;d.has_more?f(p+1):alert(i)})};f(1)

It works from anywhere in stackexchange(and many other sites), no need to be in a specific page as in @Alconja/@NateKerkhofs answers

jQuery 191

i=0;function f(p){$.get('//api.stackexchange.com/2.2/questions/featured?site=stackoverflow&page='+p,function(d){for(x in d.items)i+=d.items[x].bounty_amount;d.has_more?f(p+1):alert(i)})};f(1)

It works from anywhere in stackexchange(and many other sites), no need to be in a specific page as in @Alconja/@NateKerkhofs answers