Well, if you will allow me to include a non fast food burger, I will tell you of a hamburger I made for a bbq once.
Black truffle infused lamb hamburgers, fried in organic Italian olive oil, served on a sour dough bun with watercress tossed in walnut oil, with goat's curd spread on the inside of the top bun.
Hamburger heaven. Just bliss.
If you do it yourself, make sure to finely chop the black truffle, shape the burgers and leave to infuse for a good day in the fridge. Grill on a searing hot flatplate.
Fast food wise, we have little variety in Adelaide. There is a place in town called 'Burger It'. They do a burger called 'It'll Brie Good', which consists of organic ground beef, sliced brie cheese, plum sauce, lettuce and cheese (and something else I forgot). Really nice.