Do we see all existing galaxies?

Answer

In principle we can see all galaxies within what we call our horizon, which is the area from which light has been able to reach us since the beginning of the universe, a sphere around us with a radius of (today) about 42 billion light years.

It is difficult to say which fraction of the universe is within that horizon. A fairly authoritative theory (the inflationary universe) states that during the very earliest moments the universe expanded exponentially, with more distant regions of space moving away from us at speeds greater than the speed of light, thus disappearing from our horizon. The piece of the universe that we can see today is only a part of what may one day show itself to us, but we cannot just say what fraction it is.

Today we can see up to about 380,000 years AFTER the Big Bang. Previously, the universe was so hot that light was constantly scattered by charged particles (especially electrons), and thus no longer gives any information about where it came from. That doesn’t really matter for this question, since galaxies formed later.

There is a difference between ‘do we see?’ and ‘can we see?’. At great distances we only see the brightest galaxies. To get a complete picture, even larger telescopes are needed, on Earth and in space, and they are working on that!

In that context, when we say that a galaxy is 13.5 billion years old, it means that the light we now receive from it left there 13.5 billion years ago, not 13.7 billion years. That does NOT mean that it is now 13.5 light years away, because the universe has expanded in the meantime. When our galaxy is born, we cannot deduce from measurements of other galaxies, but must try to find out from measurements within our own galaxy. And this shows that an age of more than 10 billion years is also very plausible for us.