Suppose I have a very advanced telescope that can detect long wavelengths of light, so I can deal with extreme redshift. I am also a very quick counter, so I can get through all of the galaxies before a new one appears. Given this, can I count every galaxy in the observable universe one-by-one, and get an accurate number?
Well, you can indeed count the number of galaxies in the observable universe but not in the way you mentioned. There are about 200 billion to 2 trillion galaxies in the observable universe. Some of them you can’t distinguish as galaxies even if you use a very advanced telescope (unless it’s something like the Hubble Telescope).
But there is one way to count and then estimate the number of galaxies in the universe. You count the number of galaxies in a very small swatch of the sky and then multiply the amount by through the angle covered by the whole celestial sphere. This can work as the universe is known to be almost homogenous, that is it appears the same almost from any direction. Yet, this would give an estimate still which is in-accurate as you possibly could not count all the galaxies even in that small swatch as you can always find even more as you look deeper for which you’ll need mirrors or lenses which are multiple meters wide.