Is it possible in principle to observe all galaxies in the observable universe? Are there fundamental physical limitations that prevent the observation and cataloging of all (or almost all) galaxies in the observable universe? If there are no physical limitations, then what technological conditions are necessary to achieve such a goal?
 A: In principle, so long as the galaxy is on our past light cone and emits light, there's no fundamental reason we can't detect that light (barring "statistical accidents" like that galaxy happening to be occluded by a closer galaxy).
In practice, we are very far from being able to measure every galaxy. Here are some practical considerations that place limits on our ability to do so. Surveys typically have to trade off between sky coverage and depth. The depth of a survey is limited by the power of the telescope you are using, the amount of time you are willing to wait for one image, the angular resolution of the telescope, and the spectral range of the survey (since the intensity of the light we receive decreases as one over the square of the distance to the galaxy, the angular size of galaxies decreases with distance, and galaxies may emit in different bands and if they are far enough away will be at such a high redshift that they may move out of the observing band). If you take less time per image, you can cover a wider area, but will not be able to go as deep. You also have to worry about the Milky Way; of course it's much more difficult to observe light from galaxies that has to pass through the Milky Way than light which does not. Furthermore it matters what kind of observation you want to make. Typically you want to measure a redshift, so some kind of spectral measurement is needed. Ideally you would measure the spectrum from each galaxy and compute the redshift by looking at individual lines, but this is very time consuming. Many surveys measure photometric redshift, which is much easier that taking a full spectrum but also has a much larger uncertainty.
The LSST will be the state of the art in the next 10 years or so for galaxy surveys. It plans to get photometric redshifts for $10^{10}$ galaxies ranging from the local group to redshifts up to and even larger than 6 over $10^4$ square degrees.
A: Galaxies in the observable universe are by definition close enough that light emitted from that place at the big bang has had time to reach us. See Misconceptions About the Universe
However, stars were not all present at the beginning of the universe. Some are within the Hubble Sphere now, but weren't at the time when light from their part of the universe began reaching us.
Others are behind clouds of dust or galaxies. Their light is blocked by a dark object or obscured by a bright one.
Also, far away galaxies can be faint. No matter how good our telescopes are, there is a limit to how faint a light they can see.
For a galaxy in just the right place behind a gravitational lens. light can be focused so that we can see it, when it ordinarily would not be visible. But other galaxies can be misaligned. Their light will be dimmed.
Many galaxies have massive black holes at their center, where millions of stars have been sucked in. This is small compared to the many billions of stars in a typical galaxy. But nothing in principal prevents a galaxies where all the stars are in the black hole. We might detect this from gravitational waves, but we would not be able to see it.
