Ahhhh- a subject near and dear to my heart and also a source of sadness, as I seem to have developed a dairy allergy.
If I'm going southwestern style...pepper jack is a must. There are some other good artisanal style cheeses that also have some nice infusions to go with this kind of burger...i.e. roasted pepper, black beans, etc. Queso fresco is also a great option.
If I'm going for anything with mushrooms...I like a blend of brie/cammembert, havarti, edam, moutardier if available, a hint of some kind of appenzeller to give it a traditional Swiss finish, and maybe some maytag bleu if I want a little more kick.
If I'm going more American style...I want a better American cheese (i.e. Boar's Head from the deli case, or even Kraft select will do) with a mix of plain jack and various sharpnesses of cheddar. If I want to kick it up a bit...more of an applewood smoked cheddar and possibly even mixed in a smoked gouda. Oh yea- and MUENSTER...I LOVE muenster.
If I'm having a Cali Style or Greek burger...I like goat cheese w/herbs and maybe a dill havarti. Some of the more nutty semi-soft cheeses would also be good.
If I'm going Italian ... I want provolone, fresh mozzarella, fontina, and a nice creamy taleggio for some bite.
If I'm doing a dessert style burger like a luther...I've thought about trying more of a dessert cheese like a blueberry stilton.
OK- I've got to stop. This is killing me.