R Jaffe's Fame Game, published in 1969, is credited as the first by The Oxford English Dictionary. ('Greatest' rather than 'best' was used on that occasion. 'Biggest' appeared in 1970 and 'best' in 1972.)
I didn't think of checking there until this morning, Fred, but Partridge's Dictionary of Catch Phrases claims ' the greatest thing' started life in the USA. There, it began as a statement of appreciation of a pretty girl. Partridge reckons it has been around in British speech since about 1950, which pre-dates my earlier date by a couple of decades.