I could speak volumes on why I love a number of different composers, but I wouldn't dare try to call one of them "the greatest." Here are some different ways to look at it, from my point of view.

Most influential: J. S. Bach
Most prolific: Haydn
Most idolized: Beethoven
Most hummable: Mozart
Most adventurous: John Cage
Best orchestrators: Berlioz, Rimsky-Korsakov, Stravinsky
Best string quartets: Haydn and Shostakovich
Best symphonists: Tchaikovsky, Prokofiev, Mahler
Best ballets: Tchaikovsky, Stravinsky
Best solo piano music: Beethoven, Brahms, Debussy
Best piano concerti: Rachmaninov
Best film music: Bernard Hermann
Most intense: Penderecki
Most influential to me personally: Stravinsky