I saw a couple of good posts on this from astronomycafe.net on their ask the astronomer page. The general consensus of all the answers out there seems to be that we do not know exactly, but there are at least a hundred billion, and possibly several hundred billion. The following is quoted from Astronomy Cafe:
We do not know exactly. Within the part of the universe we can observe there seem to be at least 100 billion, but this could be an underestimate if you include dwarf galaxies that are too far away to be easily seen by even the Hubble Space Telescope.
Imagine looking at a dime about 75 feet away. The Hubble Space Telescope can represent this narrow “keyhole” stretching to the visible horizon of the universe. This “deep field image” shows hundreds of galaxies in a region only an arcminute across. What is an arcminute? Take the diameter of the full moon, divide it by 30 and that’s about an arcminute. While this is a tiny tiny view of the visible universe, it can give scientists a way to extrapolate and estimate how many galaxies there may be.
Now, even more interesting, the following answer comes from Kathy Wollard’s “How Come?” book. Our galaxy, the Milky Way is an immense spiral galaxy with about 200 billion stars. This number of stars is almost unimaginable to us, but even more astounding is that each star is often trillions of miles from its nearest neighbor star. This is how big ONE galaxy can be. Current theories are that these huge galaxies are probably the result of tens or hundreds of smaller galaxies colliding and becoming one. Of course, a “collision” in this sense actually can take millions of years. The basis of this theory is that when astronomers look far out into space, more than 2 billion light years (which is also looking 2 billion years into the past), they see more small galaxies and fewer big ones.
Imagine this: the light we see coming from a galaxy 2 billion light years away, is 2 billion years old. This is just hard to comprehend.