It's hard to say as there are so many sanctioning bodies nowadays. Different sanctioning body can has her own Champion. Actually, there are 17 divisions in professional boxing. It can be 17 Champions in each sanctioning body.
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Probably Pacquiao
WBO Welterweight champion
Ring Magazine / IBO Jr Welter champion
WBC Lightweight champion
WBC / Ring Magazine Jr lightweight champion
Ring Magazine Featherweight champion
IBF Jr Featherweight Champion
WBC Flyweight Champion
Actually Oscar Delahoya is the only true 6 weight class champion. He won MAJOR titles in all 6.
If it is most boxing championships (major and minor, regardless of division) that would be Roy Jones Jr.
IBF - Middleweight Champion
IBF- Super Middleweight Champion
WBC- Lightheavyweight Champion (3 times)
WBA Light Heavyweight Champion
WBA Super Light Heavyweight Champion
WBA - Heavyweight Champion
IBF Light Heavyweight Champion
IBO Light Heavyweight Champion
Ring Magazine - Light Heavyweight Champion
IBA - Light Heavyweight Champion
WBF - Light Heavyweight Champion
NBA - Light Heavyweight Champion
IBC - Light Heavyweight Champion
NABO - Light Heavyweight Champion (2 times)
WBC Continental Americas - Super Middleweight Champion
UBO - Intercontinental Light Heavyweight Champion
John L Sullivan
James J Corbett
Bob FitzSimmons
Jim Jeffries
Marvin Hart
Tommy Burns
Jess Willard
Jack Dempsey
Gene Tunney
Max Schmeling
Jack Sharkey
Primo Carnera
Max Baer
Jim Braddock
Rocky Marciano
Ingemar Johannson
Gerrie Coetzee
Then in recent years / today about the same again winning one portion of the belt or another including:
Vladimir & Vitali Klitschko
Sultan Ibragimov
Nikolai Valuev
There are hundreds of championship belts given in boxing. Championship belts are given from various organizations each different from the rest. It would be hard to find exactly how many have been given so far, though 354 titles were given according to an article written in August 2011.