Answer:
There is an astronomical number of planets in the universe. We are only now beginning to be able to "see" planets around other stars because of the vast distances that exist between us and them. How many there actually are is anyone's guess.
What we know about galaxies is that the ones we observe contain 200-400 billion stars each. Our solar system contains 8 planets. If we assume each star has 1 planet, and not more, and we average a galaxy to 300 billion stars we can deduce:
The Hubble Deep Field image contains 10,000 galaxies. At 2.5 arcminutes, that's roughly 1/2,000,000 of the visible universe in any direction. So if we multiply 3,000,000,000 x 10,000 x 2,000,000 we get 6 sextillion or 6,000,000,000,000,000,000,000 planets in the visible universe. It is likely much more than that.