Classical Top Five

If most will concede Beethoven, Bach, Mozart, and Brahms as " the given" top 4, who would you choose as number 5? 
Let's start a war. Toss Brahms!  Too stodgy! Go Mahler, Schubert, Stravinsky!
millercarbon -- On further thought I have to agree. Now, where's that tube of hydrocortisone to cure my illy itch?
Remove Brahms, insert Haydn.
#5... Schubert.

Later composers will be influenced by the top 5.

Sorry, but how can Mahler, Sibelius and Stravinsky be left out.  So, have to remove Haydn and probably Brahms.
WAGNER UBER ALLES! The favorite composer of that notorious Austrian with the toothbrush mustache!
I agree with tlayman18 that Haydn deserved inclusion in the Top Five. He practically invented the Classical Symphony - 104 of them, eh!
I read a book once on this very topic! J.S.Bach was at the top! And a very persuasive case indeed was made for him!
My two favorite composers, Bruckner and Mahler, definitely appear in the next five! One looking backward to Bach, the other presaging the Twentieth Century and Shostakovich!
I don't concede anything. I respect those 4 pioneers but much as I've tried to appreciate them, they quite simply bore me and there's nothing I can do about that.

1 - Stravinksy
2 - Bartok
3 - Shostakovich
4 - Holst
4 pioneers...
Kin to saying the 4 horsemen of the apocalypse set the table for the barechested q'anon shaman. 

And  Brahms is far ahead of Mahler, unless you like hearing dogs chasing their tales.
J.S. Bach is number 1.

2 and 3 are Beethoven and Schubert or Schubert and Beethoven.  Listen to Schubert's Lieder and piano sonatas then explain to me how he is not deserving of this ranking.

4 and 5 are Brahms and Haydn or Haydn and Brahms.

You can fill out a top10 with Mozart, Tchaikovsky, Mendelssohn, Schutz, and Mahler in whatever order suits you.  

I'd develop a separate list of post romantics.   Call me crazy, but Benjamin Britten might be at the top of that list for me.  Exquisitely crafted music.
I love my Benjamin Britten. He was also a heck of a fine conductor. His recordings of Mozart's Symphonies 25 & 29 on London Decca are constantly on my 'table. String tone to die for, as well.
All of the "Fifths" are great but the music I heard atop Mt. Olympus was created by Alan Hovhaness.
Doesn't matter. It's not important who wrote what, what is important is that it was written.
Mahler, and dare I say.....

Rachmaninoff, if he didn’t have to stop composing and conducting in order to make a living as the greatest all-time pianist.
And, of course, Schubert and Haydn have to be mentioned.
Mahler and Haydn.  Ditch the Brahms ...  but ... well, you know, those damn trios ...  then ... well, gotta include Dvorak, and then again there's Martinu and don't forget ... Damn! Chopin is probably the most influential of all, then ... well there's  Wagner of course ... or Cavalli ...  and Shostakovich, can't leave him out ... and ... and ...
Top five in what respects? If I had to pick the single greatest composer it would be J.S. Bach, but, in terms of influence on other composers, one could argue his son C.P.E. Bach was more influential (pioneer in early classical form) with dad sort of stuck in the baroque.  I don't think Brahms was that influential either as a neoclassicist in the romantic period.

For personal favorites, Schubert would easily be in the top five.  I would hate to have to choose between the likes of Haydn, Shostakovich, Stravinsky, Schoenberg, Verdi, Puccini, Tchaikovsky, Brahms, etc. for position outside of my top four: Bach, Beethoven, Mozart and Schubert.
@larryi I hear that comment about J.S. Bach being "stuck in the Baroque."  In a sense it is true, in that he persisted writing High Baroque after everyone else had moved on.  

The reality is that while remaining true to the baroque style for 2 decades after everyone else began to move on, that late High Baroque music went to the pinnacle of western music.  In that sense, he most certainly was not stuck anywhere.  The Matthaus passion is the greatest piece of music ever written by a human being.  If that is stuck, then give me more stuck. 

As for influence, while it is true that J.S. Bach's music was largely lost to the public until Mendelssohn reintroduced it to the public, it is not true that his work was unknown to the cognoscenti.   His influence on Beethoven is well documented as is his influence on Mendelssohn.   I'd argue that apart from Beethoven's early exposure to J.S., Beethoven would not have become the Beethoven he became in his maturity.   Beethoven didn't get the late quartets and piano sonatas from Haydn or Mozart.

For many of us, the music of J.S. Bach is the standard against which all music is judged.  C.P.E. Bach?  Not so much. 

I agree with you about Bach.  I was just raising an argument about what counts as greatness and whether influence of future composers can arguably be the main measure.  I also agree with you about the greatness of the St. Matthaus Passion, although I might still favor the B Minor Mass.  The only thing he didn't do was opera, but, his vocal works certainly show that he had what it took to do opera if he had thought it a worthy endeavor.  
The only knock against Bach and it is completely hypothetical. He composed for roughly 40 years. Mozart composed for 20. If you could switch that would you? And would it have made that much difference in our appreciation of Bach?
It would be extremely difficult to speculate what Mozart would have done with more years; I would expect really great things, but, who knows?  Based on early accomplishments, Korngold should have been a musical giant; he was good, but not that good.  Given what he had done in his short life, particularly what he did in the last year or so of his life, I would say that Schubert's premature death was the biggest loss to the world of music.
Bach, replace Mozart with Haydn, Beethoven, replace Brahms with Schubert.  5th?  Rachmaninoff. 
Arguments as to who is the greatest composer are, of course, subjective.
The composer who is the most meaningful to you is the “greatest.”
For me it’s Beethoven, then Mozart, then Bach.
Of course in the sheer volume of first rate compositions, Bach is probably the most prolific. But, to me, Beethoven, also by dint of volume, but also in the variety and scope of his masterpieces is the most meaningful.
I don’t believe anyone reached higher in inspiration than Mozart, but his short existence on this earth limited him.

How is Haydn by default, not in the top 5?
History suggests Mozart and Beethoven studied with Haydn.
He was called "papa" for a reason.

I would shoehorn Vivaldi in as well, since he was creating before all of them, the top 5 should be a top 6!

Bach can not be moved by strong  Christians , so far above all else
in service of our God.

IMO,  a sign that someone  tends to be not deeply into Classical Music is ignoring Brahms .
Putting him down is a sure sign.

I just had my left knee replaced last week so strong doses of narcotics, sleep, music, and reading. Gould's biography "Wondrous Strange" while listening to his Bach interpretations. There are times where I expected the next note to stop The Universe spinning and the doors to heaven to open. 
Try as I might, I'll always admire JS Bach more than truly love the guy. Just like Brahms, his stuff often strikes my ears as more contrived than inspired.  Either didactic or cranked out for a paycheck. It doesn't help that as the King (or whoever it was) told Mozart in the play Amadeus, there are just "too many notes." It's not that I haven't tried to give him his due, either. My record and CD shelves aren't exactly devoid of the dude. I dutifully toil through his Well Tempered Klavier on my piano.

My sympathies on your knee replacement. I had one done six months ago.
No picnic and a lot of pain.
Hang in!
Listen to a lot of Gould’s Bach.
@kenrus -- Yeah, one of my favorite records is a l'oiseau-lyre two LP edition of Handel's "Acis & Galatea" featuring an in-her-prime Joan Sutherland, but I mostly just can't get behind the composer. For me, his stuff just feels too specifically targeted to the Upper Clahhhses.  It lacks emotional power. It seems composed in a way to give an audience the okay to socialize and do business as it's being performed. 
For top 5 I would include the five who had the greatest effect on the musicians who followed after them. If you take out any of my five recommendations, most likely every single 20th century classical writers output would be impacted.
The one who stood at the epicenter of music development was Bach. His works are mathematical perfection.He is the only Western composer whose works are considered as registering on the classical musical scale among the classical Indian musicians. (Musical theory, and instrument development in India has preceeded Western music by many centuries, so it's not a little accomplishment to be acclaimed and accepted by the East.)Before Bach people have not understood the mathematical background of music so profoundly, and after Bach, they overcomplicated things, and by doing that, they also took away from the mathematical balance that Bach has achieved. Western music history in a nutshell: striving towards perfection of harmony and balance, reaching it with Bach, then loosing it again by overcomplication since Bach.
While Bach's works were lost soon after his death, many of his contemporaries were deeply affected by him - Handel, Haydn, Beethoven all were influenced by him, just as other musicians of his era.Mozart would be the second giant, with Beethoven and Haydn. Of these four, there's not doubt that they had a profound influence on Western classical music. Fifth would be.... well, hard to tell. There were so many after Beethoven and Haydn that it's impossible to pick on definitive giant who influenced everyone after him. I'd rather go back in time, to before Bach, and pick Claudio Monteverdi, a giant who made a huge turn on the wheel of music: polyphonic music, opera, and huge advances in form and melody. Without him, we would be still stuck at early Baroque, and there would not have been a Bach...  
So, I would recommend:Monteverdi,Bach,Haydn, Mozart, Beethoven. (In order of birth...)Top two, without shadow of a doubt, were Monteverdi and Bach - take them (or either of them) out of the picture and every other composers output would be drastically impacted. Contemporary music would be nothing like today.
No one mentioned Paganini. Heard the name ?
So it's Bach, Beethoven, Mozart, Paganini and Zyryab.
Jeez, let's now go with the 5 greatest rock bands...
1. Beethoven
2. Mahler
3. Stravinsky
4. Mozart
5. Tchaikovsky
6. J S Bach
Is that five already?? Doesn't matter who their influence was or when they composed, those weren't caveats.
This is such a personal choice.  If I guage this by what I actually listen to, Beethoven overwhelms this list for me.  After Beethoven, I don't think I would have a ranked order, but neither Brahms nor Tchaikovksy would be in my top 5 - probably top 10, certainly.  Mahler, Sibelius, Wagner, possibly Chopin, Haydn, Dvorak and others are top contenders.

Curious about the person who posted Holst - I have a project on my to-do list to spend a bunch of time listening to Holst past the Planets.  I have a handful of other recordings, but mostly he seems to be a one-hit wonder. I have heard of people owning 30+ recordings of the Planets and no other Holst.  
Tchaikovsky is the obvious -  however my personal top 5 are Mozart, Beethoven, Copland, Grieg and Debussy-  round out to top 8- Bach, Ravel and Tchaikovsky and lastly to 10- Vivaldi and Strauss