Up until 3 years ago I barely cropped my photos. I was always afraid to loose in quality as when image is cropped, using it in the same size you would see more artifacts and grain. In reality the image composition, the story it can tell is sometimes more important that the quality and polish. Extreme cropping is actually really interesting and you can get results you would never expect or anticipate while shooting.
There are multiple reasons you would want to play with cropping. One, is to compensate zoom that you might not have, which was my reason number one. At some point I broke my Canon 1D Mark IV (sunk in the sea) and ended up shooting on Fuji X100 for almost a year. Not having ability to change lenses and having fixed 23mm bothered me for quite a while, so I ended up cropping a lot of my photos to somewhat compensate that.
Another reason which is even more important — shift in composition. Sometimes cropping can bring up elements in your composition in a completely different way and almost tell a different story to the one that you originally shot or enhance it.
And the last reason is simply getting rid of unnecessary details that got into the shot, to keep it very clean and focused on the object.
Below you can see some of the examples where cropping the image really helped.
1. Here's an example of the photo that I took in Barcelona from the roof top. I have almost deleted this picture from my computer, as I though I would never use it. What caught my eye the very last minute are the beams of light that are coming through the mountain Tibidabo on the sunset. So I just cropped the photo in the center where the most interesting part is and got rid of dark surroundings.
2. Here's another example that I shot on Maui Island in Pe'ahi bay where surfers ride Jaws. This was the max zoom I could get using Canon - EF 600mm f/4L and you can see a lot of supporting staff and surfers on jet-skis in the frame while the wave is closing above the surfer. Cropping this image really made it much more dramatic leaving the surfer one-on-one with the wave.
3. This shot was made in Tokyo in Meiji Jingū Shrine. You can see the photographer taking a picture of newly married couple. Cropping this image in a way that I could "leave the couple alone" made it look more melancholic and lonely. Also the way couple appeared on the left of the photo looking that direction (instead of looking to the right, to the future) helped the picture a lot.
4. This picture was taken in New York from High-line Park in Chelsea. The girl jumped out of the cab and was running across the street under the rain. This crop helped me to avoid unnecessary details and all the noise around to keep focus on the cab and a girl.
5. This picture was taken from the Golden Gate Bridge of San Francisco's skyline with fixed 23mm on Fuji X100. In this example I was cropping an image to compensate zoom that I didn't have.
6. This is a view from Roosevelt Island in New York City across the East River. After fixing the perspective of the building I cropped it to get rid of it's edges. That made the photo and the building itself look more endless and brutal.