Each Girl Scout Council has the option to offer its own badges, called "Council's Own" badges (this is a link to a wiki of sorts listing many of them available). Troops, in theory, can have a Troop's Own as well, but for the most part, I've only seen council's owns. GSEM has a few, but what always fascinates me is the variety of badges available nationwide to registered scouts if only the leader knows where to look.
For instance, my juniors love archery. They love it. They keep asking when we'll go again, if we'll have a chance when we camp (in January!!). I picked up my badgebook and found that there isn't an archery badge. So I hunted online and found that the Girl Scouts of the Jersey Shore, a council in Georgia, and one in Massachusetts offer archery badges. I took a look at requirements, made them a little harder in some cases, and presented it to my girls. Three of the 19 have earned it already (they were juniors last year and had a head start) and I expect the majority to earn it by springtime.
Girl Scouts of Northern California has a self defense badge that I am tweaking a bit for my two juniors who are very involved in tae kwon do. Central Indiana has several interesting badges available, including Animals in Our Community, which my troop will most certainly earn this year; bicycling; and archaeology. There's a wide variety across the US. All these badges are worn on the front with official insignia--they are not simply participation patches. There are some brownie try-its available, too, although not as many, and interest patches for older girls.
Several times in the past year a girl has asked me "do they have a badge for...?" and I haven't known for certain. It takes some digging, but almost always I can find something that fits. There are all sorts of badges out there--nearly all that are listed at that wiki page above are open to all GSUSA scouts; some councils limit who can earn their badges, but many offer them to any scout that completes the requirements.
I wish that our council had more to offer in this regard--especially considering changes coming in the next year, it might be good to consider offering more of these options. But until that day comes, I'm happy to hunt and find options for my girls when their interests and expertise don't fit exactly into the GSUSA materials and requirements.