NOT A DAMN THING.
To me, a hamburger should consist of only two things: bread and meat. Medium rare and juicy. Anything else distracts from that wonderful charcoal-broiled ground beef taste.
And phooey, I say phooey, on those restaurants that won't cook a burger less than medium or medium-well. One obnoxious waiter in Ohio told me that "It's illegal in the United States to cook ground beef less than medium well." Nobody ever told that to my local diner in New Jersey, bucko.
Now that I've had my rant, I will admit that the diner in town has a couple of "special" burgers that I occasionally enjoy: a double burger with mushroom gravy (messy, but fun) and a burger topped with bacon, lettuce, and tomato (hold the mayonnaise or I'll serve you a knuckle sandwich).
But I will never, ever, ever, eat cheese on my hamburgers.