Hello, I'm new here. This is an interesting topic, but as someone who has done quite some research in this field, I must say there are quite some errors here.
Among composers, Mozart, Bach and Beethoven have always been the most popular composers. If you were to hold a poll among composers, probably Mozart would end as number 1, with Beethoven a close second. Here some favorite composers of great composers:
Haydn considered Mozart the greatest composer he knew. He said he learned the most from C.Ph.E Bach. He considered Beethoven very talented, but his music awkward as well.
Beethoven considered Mozart the greatest composer, till he learned to know Händel better, than Händel became his favorite. He also admired Haydn and Bach.
Von Weber’s favorite composer was Mozart. He wasn’t too fond of Beethoven.
Rossini’s idol was Mozart. He also valued Bach very highly, as well as Haydn and Beethoven (though he preferred the early Beethoven).
Schubert idolized Mozart. In his youth he had some reservations about Beethoven, but later in his life that completely changed. He also worshipped Haydn and Hummel.
Berlioz’s idols were Beethoven and Gluck. He was ambigious about Mozart, whom he called the greatest musician in the world, but of whom he could also be critical. Haydn he didn’t consider very important. He even strongly disliked Bach and Händel.
Schumann particularly admired Beethoven and Bach, as well as Mozart and Schubert. Mendelssohn he called “The Mozart of the 19th century”, and he also championed Chopin. He was less positive about Wagner. He didn’t see the great value of Haydn.
In his youth, Mendelssohn considered Beethoven the greatest composer. In later life he moved a little more in the direction of Bach and Mozart. He also worshipped Händel and Haydn. He didn’t share the same enthousiasm for his contemporaries.
Liszt’s great hero was Beethoven, followed by Schubert. He revered Bach as well, mainly later in his life I believe. He also admired Mozart and Von Weber.
In opposite to what people say here, Wagner was more modest than one would expect. He said he wasn’t even allowed to tie Beethoven’s shoe laces, for example. Wagner’s remarks are a bit confusing. Normally he’s most associated with Beethoven, but at the same time he said of Mozart that “the most tremendous genius raised Mozart above all masters, in all centuries and in all the arts”, and about Bach: “the greatest miracle in all music”, and about Liszt that he was the greatest musician in the world. His disapproval of Mendelssohn is notorious, but don’t forget he once sent his first symphony to Mendelssohn, who lost it (which may explain Wagner’s later attitude to Mendelssohn). But he called Mendelssohn nevertheless the greatest purely musical talent since Mozart. Wagner made also very cautious remarks about Schumann.
Chopin’s heroes were Mozart and Bach. He also liked Haydn and Hummel. His opinions of Mendelssohn are contradicting. He was full of disdain for Berlioz. In opposite to what is often said, he didn’t dislike Beethoven at all, though it was clear Chopin was of a different temper.
Brahms’ favorite composer was not Beethoven, but Mozart, though he deeply admired Beethoven as well. Next to Mozart, Bach was his favorite. He also worshipped Schubert, Haydn, Mendelssohn, Schumann and Dvorak, and even Wagner.
Mozart was also Tchaikovsky’s idol. He admired but didn’t like Beethoven, considered Bach good, but no genius, and disliked Brahms fullheartedly, though he admired him for his sincerity. He preferred Grieg instead.
Also Grieg considered Mozart unattainable, even compared to Bach, Beethoven and Wagner. Chopin is often mentioned as Grieg’s favorite composer, but I found no evidence for that, I think it’s due to the fact they both wrote mainly for the piano. Grieg loved Verdi, Schumann, Mendelssohn and Dvorak as well. He saw Richard Strauss’ great gifts, but didn’t like what he did with it.
Bruckner’s idol was Wagner. He was extremely humile about his own achievements compared to other composers, mainly to Wagner.
Dvorak considered Mozart the greatest of all composers. He admired Beethoven and Schubert very much, as well as Brahms and Wagner.
Saint-Saëns main heroes were Mozart, Bach and Beethoven, Haydn as well. He admired Wagner as well, and had some reservations about Berlioz, though he recognized his genius as well.
The principal heroes of Mahler were Wagner and Beethoven. He also deeply admired Mozart (he died after murmuring “Mozartl”, little Mozart) and Bach.
The favorite composer of Debussy was Chopin. Bach was a god to him, as was Mozart. He considered Wagner and Beethoven geniuses, but could also be very critical of them. He was ambigious about Liszt and Berlioz, and rather negative about Mendelssohn and Schubert. He was inspired by the Russians, but felt little sympathy for Tchaikovsky.
Richard Strauss’s god was Mozart. He admired Schubert, Haydn, Wagner, Beethoven and Bach as well. He disliked atonal composers.
Schoenberg named Mozart and Bach as his principal teachers, the latter he considered the greatest of them all. Next came Beethoven and Brahms. His tastes were mainly Teutonic, he also had a high opinion of Mendelssohn and Wagner. He could be caustic about Stravinsky.
From his early life on, Stravinsky championed Mozart and Tchaikovsky, while he disliked Beethoven. Later he learned to appreciate Beethoven, and he revered Bach. He once named Schubert as his favorite composer. He had less positive things to say about Wagner, Berlioz, was partial to Brahms, and considered Schubert far more interesting than Schumann. Besides them he loved Rossini, Mendelssohn and Von Weber.
Prokofiev considered, maybe surprisingly, Haydn as his favorite composer. Tchaikovsky he admired as well.
Bartok’s early love was Beethoven. Other composers he admired were Bach, Debussy, Mozart, Richard Strauss. He disliked Shostakovich.
The favorites of Sibelius were Bach, Mozart and Beethoven. He also loved Mendelssohn, whom he considered with Mozart a greater master of orchestration that Strauss or Stravinsky.
Ravel regarded Mozart as the most perfect of all composers. He could be rather harsh about Beethoven, though he didn´t hate him, as is suggested here. Like Debussy, he loved the Russians (but not Tchaikovsky), as well as Schubert and Mendelssohn.
Vaughan Williams had Bach on top of his list. He had mixed feelings about Beethoven and preferred Haydn to Mozart.
Hindemith considered Bach the greatest composer and deeply admired Mozart.
Shostakovich´s tastes were eclectic, as he put it, he liked everything from “Bach to Offenbach”. He was fond of the classics, and in his youth he considered Mahler the greatest composer of all. Among Russians, he venerated Moussorgsky and Tchaikovsky, and considered Stravinsky the greatest composer of the 20th century.
Britten’s idol was Mozart. He loved Schubert a lot as well, and admired Bach. In later life, he developped a disliked for Brahms and Beethoven.
Messiaen’s favorites were Debussy, Wagner, Mozart and Berlioz. Except for Mozart, he considered the classics and the baroque composers as composers who knew nothing about rhythm.
Boulez’s favorite composer I think was Webern.
Ligeti admired Brahms, considered Mozart’s operas the greatest of all, and Cosi Fan Tutte the most beautiful music there was, and saw the operas of Verdi as close to the level of Mozart’s.
Did I miss something yet? ;-)