The largest muscle groups in the human body would be the glutes and quads. Exercise that work those body parts will get you a good number of calories burned per minute. You want to incorporate as many muscle groups as possible to burn the highest number of calories:
Versa-climbers, which work the whole body, burn the most calories. They aren't cheap and might not be easy to find at a gym. Nordic Ski machines (or actually skiing on snow) and rowing machines also burn massive calories. Because your arms, legs and booty are all working, these will get it done.
Swimming, step aerobics, stair climbing, running... all good choices for burning calories. But more important is - what do you enjoy doing?
Although I love the idea of what a versa-climber does for my body, I hate
being on it. I'd rather be fat!
On the other hand, I love hiking. Burns fewer calories, but I can keep at it much longer and enjoy doing it - so hiking is a better choice for me.