Go to the deli and see if you can get some Hoffman’s Super Sharp Cheddar. Best cheddar ever! You don’t say where you live, but I know Giant Eagle and Hannaford carries it.
Pepper jack and/or Gouda…spiced or smoked cheeses in general are excellent, had a horseradish Gouda cheese once that was awesome! Don’t agree with mozzarella, though…too mild, taste will get lost among the rest of the burger. Provolone? Okay, but there are better choices. Blue cheese, eh…only okay if you don’t overdo it.