The biggest problem would be reproducing the captured image.
It's not outside the realm of technology to create an image sensor and configuration that would capture an extremely wide range of brightness levels in a single image. In the end it's just a matter of photon-counting, which is a technology that does scale to the necessary levels. Current cameras primarily use exposure settings to modulate the amount of brightness that the sensor sees, though more of this work could be done in the sensor, perhaps resulting in greater error noise, but you could certainly get a wider range out of a photo sensor than what is currently available on the market.
But the problem is this: once you have that picture, what do you do with it? Even high-end displays still use 24-bit color, meaning only 256 shades per color channel allowed. Current printers are similarly limited, if not more so. So nothing could actually be done with such an image without some processing first to reduce the range down to what existing cameras produce.
You've probably seen this problem before: most current RAW formats already store a wider range than can be reproduced, and the color range already has to be compressed or clipped before you can look at the picture. Adding even more range to the RAW output would just be more of the same. The camera would likely be dramatically more expensive but the pictures wouldn't be significantly better because you still have to chop the range down to 24-bit color before you can look at it.
Still, perhaps with the right software and the right kind of user, you may be able to get something wonderful out of it. It'd probably be not very unlike current HDR photography, but you wouldn't have to snap multiple images.