No-one is really sure exactly how many there are. Many millions probably. One thing that might be useful in answering this is a 2 year survey (the 2dF Galaxy Survey) which has just finished.
So far they have surveyed 100, 000 galaxies to make a 3-D map of the universe, but their goal is 250,000! And that’s certainly not all of them that are out there. Look at the 2dF website for more on this. There are 4 million names in the NASA Extragalactic database! (NED) Although this is probably not everything either.
It is estimated that there are as many as 200 billion galaxies in the observable universe, but we aren’t able to see all of them yet as our telescopes are not big enough. This number is interesting because it is similar in magnitude to the number of stars estimated to be in our Galaxy.