Ok - I'm not even going to attempt to guess how many nights I've stayed at Hyatts (edited 500?) but this is the list of the ones I've stayed at (IIRC) - 76 total properties - the most number of nights is probably at the former Park Hyatt San Francisco followed by Buffalo and my mattress run property Cincinnati.

Summerfield (3)
Mt Laurel
Plymouth Meeting (Philadelphia)
Herndon

Hyatt Place (7)
Scottsdale Old Town
Phoenix - North
Nashville Airport
Cincinnati Airport
Cincinnati Blue Ash
Las Vegas
Dallas/Plano

Hyatt Regency (37)
Scottsdale (Gainey Ranch)
Century Plaza
San Francisco
San Francisco Fisherman's Wharf
San Francisco Airport
Grand Champions
Irvine
Huntington Beach
Newport Beach
Denver Tech Center
Grand Cypress (Orlando)
Pier 66 (Fort Lauderdale)
Atlanta
Suites Perimeter (Atlanta)
Waikiki
Chicago
Chicago O'Hare
Lexington
Louisville
Cambridge
St Louis (Former Hyatt)
Albuquerque
Tamaya Resort
Buffalo (actually "lived" here for 2 months in 1991 when moving back to Buffalo then)
Rochester
Cincinnati (my Mattress Run property - actually "was" on top 50 customer list)
Pittsburgh Airport
Dallas
DFW
Houston
Reston
Bellevue (WA)
Montreal
Vancouver
Auckland (Former Hyatt)
Mainz
London Churchill

Grand Hyatt (11)
San Francisco
San Diego
Denver
Tampa Bay
Kauai
New York
Seattle
Hong Kong
Singapore
Bangkok
Dubai

Andaz (2)
Los Angeles (West Hollywood)
Fifth Avenue

Park Hyatt (10)
Chicago
Washington DC
Toronto
Sydney
Tokyo
Paris Vendome
Istanbul
Los Angeles (Former Hyatt)
San Francisco (Former Hyatt - my all time favorite Hyatt - probably 90 stays)
New York Stanhope (Former Hyatt)

Others (6)
Highlands Inn (former Park Hyatt Carmel)
Victor (Former Hyatt)
Grand Victoria Casino (Former Hyatt)
Capitol Square (Columbus) (Former Hyatt)
Bellevue (former Park Hyatt Philadelphia)
Olive 8

Edit: I compiled this list from the "Property Index" sticky - it was easier that way to figure out which ones I've stayed at rather than from my GP account.

Another Edit: Realized I left Century Plaza & HP Dallas/Plano off.