According to the data at the unofficial Wiki, a galaxy is composed of 4.2 billion regions (slightly less than 2^32). Each region can theoretically contain 4096 star systems (2^12), but in practice, far fewer appear on the map*. 350 systems per region appears to be the average. Each star system contains between two and six bodies (planets + moons); four planets appears to be the average.
Multiply it all out, and you get roughly 6 quadrilliontrillion planets per galaxy.
*Since the game is procedurally generated, the non-mapped star systems can be visited through savegame editing or bugs, raising some interesting philosophical questions about what it means for something to exist.