Green’s Dictionary of Slang

greatest, the adj.

(US) excellent, fantastic, a general term of approval.

[US]Rosenthal & Zachary Jazzways 56: ‘Duke’s the greatest’ is certainly the easiest cliché tossed around in swing circles.
[US]Esquire Nov. 131: Jazz musicians and enthusiasts thrive on hyperbole, of course; if anything is good, it is ‘the greatest’.
[US]P. Crump Burn, Killer, Burn! 336: Take me in your arms and kiss me. That would be the greatest!
[US]W. Murray Sweet Ride 98: ‘Was it that good?’ ‘The greatest. You should have been there.’.
[Aus]B. Humphries Barry McKenzie [comic strip] in Complete Barry McKenzie (1988) 120: Why sir, this tour is just the greatest.