Cover 3 and Old School Grill. Cover 3 was one of the Austin places in Texas Monthly's Top 50 burgers and Old School Grill is a local place that has a similar style burger, but it just tastes a little more "homemade".
OSG is probably preferable to those who want a simple burger whereas the Cover 3 "Hickory" has the cheese/bacon/bbq that some people prefer.
Edit: after trying Casino El Camino last week I'd have to put that on top of the list. Very flavorful meat, unique sweet/grilled bun, good variety of toppings.
Edit #2: P Terry's is a very solid option for a <$5 burger – way better than anything else in that category.