The record currently stands at 31 championships. The record is held by recently retired superstar Edge. The first title he won in the WWE was the Intercontinental Championship in June, 1999. Since then he has become the fourteenth WWE Triple Crown winner, and is only one of two men to have held every currently active WWE title. The other is Kurt Angle.
Here is the breakdown of the championships that Edge has won:
World Heavyweight Championship – 7 times
WWE Championship – 4 times
WWE Intercontinental Championship – 5 times
WCW United States Championship – 1 time
WWF/E World Tag Team Championship – 12 times
with Christian (7), Chris Benoit (2), Hulk Hogan (1), Randy Orton (1), and Chris Jericho (1)
WWE Tag Team Championship – 2 times
with Rey Mysterio (1), and Chris Jericho (1)