The annual Mardi Gras in New Orleans, Louisiana, has been called the “greatest free show on earth.”  “Biggest free show on earth” has been cited in print since at least 1937, and “greatest free show on earth” has been cited in print since at least 1938. The “Greatest Show on Earth” is a trademark slogan (first used in the 19th century) for Ringling Bros. and Barnum & Bailey Circus.
 
The Mardi Gras organizers tried to de-emphasize the slogan—and the word “free”— in the 1970s, encouraging visitors to cooperate by patronizing New Orleans businesses.
 
   
Wikipedia: Mardi Gras in New Orleans
The holiday of Mardi Gras is celebrated in Southern Louisiana, including the city of New Orleans. Celebrations are concentrated for about two weeks before and through Shrove Tuesday, the day before Ash Wednesday (the start of lent in the Western Christian tradition). Usually there is one major parade each day (weather permitting); many days have several large parades. The largest and most elaborate parades take place the last five days of the Mardi Gras season. In the final week, many events occur throughout New Orleans and surrounding communities, including parades and balls (some of them masquerade balls).
     
10 February 1937, The Delta Star (Greenville, MS), “Hangovers In New Orleans Today” (UP), pg. 1, col. 8:
New Orleans’ historic Mardi Gras—“Biggest free show on earth”—packed streets for miles with a seething turmoil of revelry, color and sound.
       
Google Books
The Highway Traveler
Volumes 10-13
1938
Pg. 43:
Mardi Gras Week needs little mention. Growing in popularity each year, this outstanding event in the city’s annual list of promotional stunts continues to draw people from every section of the United States. Termed by visitors’as the “greatest free show on earth,” Mardi Gras Is a period when more than half a million human beings discard the clutters of convention and express their festive feelings before the Lenten period of fasting.
 
29 January 1940, McComb (MS) Daily Enterprise, pg. 5, col. 6:
NEW ORLEANS, LA.—Final plans are being made here for Mardi Gras—the “greatest free show on earth.”
 
23 February 1940, Medford (OR) Mail Tribune, “Society and Clubs” by Clara Mary Davis, pg. 7, col. 1:
Allens Write of Recent Mardi Gras
According to word received in Medford today from Mr. and Mrs. E. T. Allen who are enjoying an extended vacation over most of the United States, they are now in New Orleans where they attended the recent Mardi Gras. The Allens wrote to friends, “Orleanians claim the Mardi Gras is the greatest free show on earth and I do not think they are far wrong.”
 
29 January 1941, Chicago (IL) Daily Tribune, “Gay Mardi Gras to Banish Gloom in New Orleans” by Frederic Babcock, Tribune Travelers’ Guide, pg. 13, col. 2:
Barnum and Bailey used to call their circus the greatest show on earth. The Mardi gras has been termed the greatest free show on earth. Certainly it is the oldest, as well as the most famous, affair of its kind in this country.
 
23 November 1941, New York (NY) Herald Tribune, “Mardi Gras Celebration Attracts Throngs to New Orleans in Winter: Week of Spectacular Festivities to Start Feb. 10 Sports Carnival Featured in December” by Richard W. Dunlap, pg. D11, col. 1:
The Mardi Gras celebration is, of course, the occasion for hundreds of thousands of visitors from every part of the country to pay their respects to New Orleans, enjoy its abundant hospitality and witness what has been properly described as the “greatest free show on earth.”
 
OCLC WorldCat record
Mardi Gras in New Orleans 1965 : greatest free show on earth! Featuring the greatest bands in Dixie!.
Publisher: [New Orleans, La.] : [Publishing Corporation?], [1965]
Edition/Format:   Print book : English : 3rd annual ed
 
OCLC WorldCat record
Mardi Gras in New Orleans 1966 : greatest free show on earth! Featuring the greatest bands in Dixie!.
Publisher: New Orleans, La. : Publishing Corporation?, ©1966.
Edition/Format:   Print book : English : 4th annual ed
 
OCLC WorldCat record
Mardi gras in New Orleans ‘69 : greatest free show on earth
Author: Thomas J Lupo
Publisher: New Orleans : Pub. Corp., Thomas J. Lupo, ©1969.
Edition/Format:   Print book : English : 7th annual ed
   
OCLC WorldCat record
New Orleans on parade : tourism and the transformation of the Crescent City
Author: Jonathan Mark Souther
Publisher: Baton Rouge : Louisiana State University Press, 2014.
Series: Making the modern South.
Notes: Preface to the Paperback Edition—A “New” New Orleans—Preservation and Profit in the French Quarter—Into the Big League—Making “The Birthplace of Jazz”—Selling “The Greatest Free Show on Earth”—“Creole Disneyland”—“A City on Parade.”
Google Books
Pg. 151:
Among the panel’s recommendations was an initiative to reshape Mardi Gras as a community event. Following the panel’s report, in 1973 the tourist commission officially abandoned its slogan “The Greatest Free Show on Earth.” In its place the commission published a new pamphlet, “The Real Mardi Gras,” which informed would-be revelers: “Mardi Gras has been called ‘the greatest free show on earth.’ It is the greatest, but it’s not free. It’s paid for by private organizations. Come and enjoy our fine restaurants, our entertainment, our shops, and our Mardi Gras. But please come as a paying guest as you would to any other great city or event.”
   
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Mardi Gras World - Behind The Scenes of the Greatest Free Show on Earth
4:33 PM - 20 Apr 2018