I checked the TAC reprint of Beck's Triples, Algebras, and Cohomology from 1967. It is evident from the discussion at pag. 8, before Thm. 1 that tripleability was not presented by Beck before his thesis (1967). There, Beck promises a paper "to appear", whose intended title was The tripleableness theorems. The TAC reprint has an editor's note in the bibliography commenting on this paper.
Editors’ note: To our knowledge, this has not appeared. Beck’s tripleableness theorems have been exposed in M. Barr and C. Wells, Toposes, Triples and Theories. Springer-Verlag, Berlin, Heidelberg, New York, 1984 as well as other places.
My impression is thus that Beck was writing the paper while working on his thesis in 1966. When people refer to this forgotten manuscript they are actually referring to the discussion before Thm 1, a draft that maybe has circulated in a very sketchy form among very selected people and was probably never finished.