I read greatest as largest ... sort of assuming which genre has the most songs.

But then that's just me slagging off slang.

Taking "great" at it's popular definition... I would say Blues is the most influential. Not that I listen to much (usually only on the weekends). It just seems to have touched so many other genre. One can pick out blues elements in so much music dating from the 1920s forward. Probably because it is musically amazing, composers can't help but pinch and adapt bits. Jazz, Swing, Gospel Blues(spiritual with a blues motif), Rockabilly (bluegrass and blues), Modern Folk (20th century) Country Western, Rock and Roll(50s forward including many sub genre), numerous world fusions, etc.

So it is really "great" (translate - fortunate) that we had Blues to bouy up and in some cases give birth to much of the music we love.