There's a Japanese fast-food chain that's very popular here in Singapore. It's called Mos Burger, so you can guess what they traffic in. You can get a regular hamburger or cheeseburger there, and I think even a version of Fillet-O-Fish, but there are some differences.
For one, the burgers are a little smaller in circumference, but the patties are thicker (I don't know whether the beef is better, since I had a different sandwich). The traditional ones are also topped with a thick slice of beefsteak tomato, which I think is a nice touch. You can have ketchup or chili sauce with your burger.

Also, they have a large line of non-Western burger-ish items. I chose a burger that was essentially Korean bulgogi in handheld form -- strips of slightly sweet (barbecued? not sure) beef, with compressed rice patties forming the bun, and a leaf of lettuce to top things off. It was pretty tasty, especially for fast food, but it turns out that bulgogi-style beef is a little hard to handle without utensils of some kind, so I had a bit of trouble keeping the whole thing contained without the rice patties falling apart. All in all, though, I'd try it again, maybe with a different choice of filling.