The Hubble Deep Field image was an exposure of 10 days. The length was required in order to capture enough photons from the dimmer distant galaxies to be detected by the telescope's CCD.

If you arranged for an exposure which is on order of magnitudes longer in duration, would so many photons eventually interact with the CCD to create an image that would essentially just be a bright rectangle with every pixel illuminated?

IE instead of having black background with a few pixels illuminated by distant galaxies, have the entire field of view illuminated by the billions of galaxies which would fill in the gaps between other brighter galaxies?