As of the end of the 2007 season, there were 3 AAA leagues and 46 AAA teams:
1) International League
Buffalo Bisons
Charlotte Knights
Columbus Clippers
Durham Bulls
Indianapolis Indians
Lehigh Valley IronPigs
Louisville Bats
Norfolk Tides
Pawtucket Red Sox
Richmond Braves
Rochester Red Wings
Scranton/Wilkes-Barre Yankees
Syracuse Chiefs
Toledo Mud Hens
2) Pacific Coast League
Albuquerque Isotopes
Colorado Springs Sky Sox
Fresno Grizzlies
Iowa Cubs
Las Vegas 51s
Memphis Redbirds
Nashville Sounds
New Orleans Zephyrs
Oklahoma RedHawks
Omaha Royals
Portland Beavers
Round Rock Express
Sacramento River Cats
Salt Lake Bees
Tacoma Rainiers
Tucson Sidewinders
3) Mexican League
Acereros de Monclova
Diablos Rojos del México
Guerreros de Oaxaca
Leones de Yucatán
Olmecas de Tabasco
Pericos de Puebla
Petroleros de Poza Rica
Piratas de Campeche
Potros de Tijuana
Rieleros de Aguascalientes
Rojos del Águila de Veracruz
Saraperos de Saltillo
Sultanes de Monterrey
Tigres de México
Tuneros de San Luis Potosí
Vaqueros Laguna
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The Mexican League does NOT count as a AAA league. It's generally considered to be of AAA-level talent, but the phrase AAA specifically refers only to the 30 teams affiliated with a major league club, one apiece.
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There are currently 30 teams.
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There were 30 teams in Major League Baseball in 2005.
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