Cropping is traditionally used to tighten a composition: to remove extraneous detail and direct the viewer's eye to that part of the image which the photographer wishes to emphasize. Unfortunately, this can result in static, almost claustrophobic images, because the composition becomes so tight and rigid.
Consider cropping for another purpose: to ADD space to a photograph (rather than SUBTRACTING space from it.) The resultant image will be much more fluid... and less expected.
Here's an example:
The original photograph shows the overall architecture of the ferris wheel (although it doesn't portray its scale as well as if there had been passengers on it at the time):
However, the photo does not communicate what drew my eye to the wheel to begin with: its center hub and the spokes. Here, cropping tightly draws the reader's eye to those features:
However, the cropped image, because it is so rigid, fails to express any kind of movement, which is why people go on the rides to begin with. This could have been solved by shooting a slower shutter speed, to blur motion. (Alternatively, motion could have been added in Photoshop, although I usually don't like to do that.) Moreover, the image fails to convey the feeling of freedom one has sitting in a car, surrounded by blue sky. Here's another cropping of the same picture, which better expresses that feeling of "openness" while still retaining architectural interest:
This image demonstrates that while it seems counter-intuitive to make a photo more open by tightly editing it, sometimes that's exactly what is needed.
Not convinced yet? Here's another example of adding space and openness to an already tightly cropped photo by coming in even closer:
Before:
and after:
