How many different galaxies are there?
Roughly several hundred billion.
The answer isn't well defined, because as we look harder (use more powerful telescopes) we see more. There is presumably a limit to this, as the first galaxies we can possibly see are those whose light has only just reached us some 13.8 billion years after the Big Bang. It also took some time for enough matter to collect (lots of stars) to form the first galaxies, though less than one might expect.
Another really mind-troubling fact is that only 80 years ago, when my grandfather was a young man, we thought there was only one, our own Milky Way galaxy, and that this was the whole universe. Truly, an expansion of consciousness in a astonishingly short time.