The Big 3 are indeed in a class of their own and you can make a strong case for each. Others may have their defenders (especially the Wagnerians) but none of 'em can match this trio in depth AND width. Let's do the primers:
Bach - The most perfect synthesis of emotion and intellect in music. His miraculous command of counterpoint remains unchallenged to this day (even Wolfgang struggled with it at times), and at his most inscrutable he has been matched only by Beethoven in his great successor's late piano sonatas and string quartets. And he's a terrific tunesmith to boot. In my book there's Bach and there's everyone else.
Mozart - The greatest natural genius in the history of Western music, if not the greatest natural genius period. He happens to be my least favorite of the Big 3 as I find most of his oeuvre too polite, but
Don Giovanni, the sublime closing pages of
The Marriage of Figaro and Sarastro's high arias in
The Magic Flute stand supreme as pinnacles of musical theater, and the legendary unfinished Requiem remains the most bone-chillingly frightening music ever written. Salieri (the fictional one, for the record) had every reason to curse the Almighty for endowing this heavenly creature with so much talent and yet taking him away from us all too soon.
Beethoven - The most widely influential, daringly innovative and intensely personal musician in history. He wasn't as gifted as Bach and (as Ludwig himself painfully admitted) Mozart and didn't share the duo's natural ear for the human voice (the "Ode to Joy" finale of his Ninth Symphony being the most infamous example), but nobody has left a deeper imprint on his music which by turns is as tempestuous, serene, violent, peaceful, impassioned, withdrawn, outgoing and enigmatic as humanly possible. The man was indeed greater than the myth.
As for the rest I actually posted a
tentative ranking of the top 30 earlier this year but here it is again, expanded to 50 names with a few changes to the original top 30:
The Kings of Kings
1) Bach, J. S.
2) Mozart
3) Beethoven
The Immortals
4) Handel
5) Schubert
6) Brahms
7) Wagner
8) Haydn
9) Tchaikovsky
10) Schumann
The Übermenschen
11) Stravinsky
12) Verdi
13) Dvořák
14) Chopin
15) Mahler
16) Liszt
17) Mendelssohn
18) Prokofiev
19) Strauss, R.
20) Berlioz
The Royals
21) Debussy
22) Shostakovich
23) Monteverdi
24) Palestrina
25) Bartók
26) Puccini
27) Sibelius
28) Vivaldi
29) Bruckner
30) Purcell
The Wizards
31) Ravel
32) Josquin
33) Fauré
34) Rossini
35) Telemann
36) Britten
37) Bizet
38) Rameau
39) Rachmaninoff
40) Donizzeti
The Grandmasters
41) Hindemith
42) Gluck
43) Saint-Saëns
44) Schoenberg
45) Vaughan Williams
46) Corelli
47) Franck
48) Smetana
49) Couperin
50) Mussorgsky
And these are the ten composers who missed the cut but could easily be swapped with some of the above. Let's call them the
Nearers, in order of birth only (since we're entering relatively obscure territory I'm including their full name and also their DOB and country of origin in parentheses):
Machaut, Guillaume de (c. 1300–1377, France)
Bach, Carl Philipp Emanuel (1714–1788, Germany)
Weber, Carl Maria von (1786–1826, Germany)
Grieg, Edvard (1843–1907, Norway)
Borodin, Alexander (1833–1887, Russia)
Janáček, Leoš (1854–1928, Czech Republic)
Elgar, Edward (1857–1934, England)
Ives, Charles (1874–1954, USA)
Villa-Lobos, Heitor (1887–1959, Brazil)
Copland, Aaron (1900–1990, USA)
Finally the honorable mentions (130 and counting), grouped by period/style for ease of reference:
Medieval (3)
Hildegard von Bingen (1098–1179, Germany)
Léonin (fl. 1150s–c. 1201, likely France)
Pérotin (c. 1155/60–c. 1200/05, France)
Renaissance (14)
Dunstable, John (c. 1390–1453, England)
Dufay, Guillaume (1397–1474, Belgium/France)
Binchois, Gilles (c. 1400–1460, Belgium)
Ockeghem, Johannes (c. 1410–1497, Belgium/France)
Tallis, Thomas (c. 1505–1585, England)
Lassus, Orlande de (c. 1532–1594, Belgium/Germany)
Byrd, William (c. 1539/40–1623, England)
Victoria, Tomás Luis de (c. 1548–1611, Spain)
Marenzio, Luca (1553 or 1554–1599, Italy)
Gabrieli, Giovanni (c. 1554/1557–1612, Italy)
Dowland, John (1563–1626, England)
Gesualdo, Carlo (1566–1613, Italy)
Praetorius, Michael (1571–1621, Germany)
Gibbons, Orlando (1583–1625, England)
Baroque (22)
Sweelinck, Jan Pieterszoon (1562–1621, Netherlands)
Allegri, Gregorio (c. 1582–1652, Italy)
Frescobaldi, Girolamo (1583–1643, Italy)
Schütz, Heinrich (1585–1672, Germany)
Cavalli, Francesco (1602–1676, Italy)
Lawes, William (1602–1645, England)
D'Anglebert, Jean-Henri (1629–1691, France)
Lully, Jean-Baptiste (1632–1687, France)
Buxtehude, Dieterich (c. 1637/39–1707, Denmark/Germany)
Charpentier, Marc-Antoine (1643–1704, France)
Biber, Heinrich Ignaz Franz (1644–1704, Bohemia/Austria)
Pachelbel, Johann (1653–1706, Germany)
Marais, Marin (1656–1728, France)
Scarlatti, Alessandro (1660–1725, Italy)
Albinoni, Tomaso (1671–1751, Italy)
Zelenka, Jan Dismas (1679–1745, Czech Republic/Germany)
Scarlatti, Domenico (1685–1757, Italy)
Porpora, Nicola (1686–1768, Italy)
Tartini, Giuseppe (1692–1770, Italy)
Locatelli, Pietro (1695–1764, Italy/Netherlands)
Hasse, Johann Adolph (1699–1783, Germany/Italy)
Pergolesi, Giovanni Battista (1710–1736, Italy)
Classical (6)
Soler, Antonio (1729–1783, Spain)
Boccherini, Luigi (1743–1805, Italy)
Clementi, Muzio (1752–1832, Italy/England)
Dussek, Jan Ladislav (1760–1812, Czech Republic/France/England)
Cherubini, Luigi (1760–1842, Italy/France)
Hummel, Johann Nepomuk (1778–1837)
Romantic (54)
Sor, Fernando (1778–1839, Spain/France)
Field, John (1782–1837, Ireland)
Paganini, Niccolò (1782–1840, Italy)
Spohr, Louis (1784–1859, Germany)
Meyerbeer, Giacomo (1791–1864, Germany)
Bellini, Vincenzo (1801–1835, Italy)
Adam, Adolphe (1803–1856, France)
Glinka, Mikhail (1804–1857, Russia)
Alkan, Charles-Valentin (1813–1888, France)
Gounod, Charles (1818–1893, France)
Offenbach, Jacques (1819–1880, Germany/France)
Lalo, Édouard (1823–1892, France)
Gottschalk, Louis Moreau (1829–1869, USA)
Ponchielli, Amilcare (1834–1886, Italy)
Delibes, Léo (1836–1891, France)
Balakirev, Mily (1837–1910, Russia)
Bruch, Max (1838–1920, Germany)
Massenet, Jules (1842–1912, France)
Rimsky-Korsakov, Nikolai (1844–1908, Russia)
Strauss II, Johann (1825–1899, Austria)
Wieniawski, Henryk (1835–1880, Poland)
Chabrier, Emmanuel (1841–1894, France)
Sullivan, Arthur (1842–1900, England)
Widor, Charles-Marie (1844–1937, France)
Sarasate, Pablo de (1844–1908, Spain)
Duparc, Henri (1848–1933, France)
Parry, Hubert (1848–1918, England)
Tárrega, Francisco (1852–1909, Spain)
Humperdinck, Engelbert (1854–1921, Germany)
Sousa, John Philip (1854–1932, USA)
Chausson, Ernest (1855–1899, France)
Ruggiero, Leoncavallo (1857–1919, Italy)
Ysaÿe, Eugène (1858–1931, Belgium)
Wolf, Hugo (1860–1903, Slovenia/Austria)
Albéniz, Isaac (1860–1909, Spain)
MacDowell, Edward (1860–1908, USA)
Delius, Frederick (1862–1934, England)
Mascagni, Pietro (1863–1945, Italy)
Nielsen, Carl (1865–1931, Denmark)
Glazunov, Alexander (1865–1936, Russia)
Dukas, Paul (1865–1935, France)
Busoni, Ferruccio (1866–1924, Italy)
Satie, Erik (1866–1925, France)
Granados, Enrique (1867–1916, Spain)
Joplin, Scott (c. 1867/68–1917, USA)
Scriabin, Alexander (1872–1915, Russia)
Reger, Max (1873–1916, Germany)
Suk, Josef (1874–1935, Czech Republic)
Holst, Gustav (1874–1934, England)
Kreisler, Fritz (1875–1962, Austria)
Falla, Manuel de (1876–1946, Spain)
Dohnányi, Ernst von (1877–1960, Hungary)
Respighi, Ottorino (1879–1936, Italy)
Medtner, Nikolai (1880–1951, Russia)
Modernist/
Early 20th-Century (21)
Enescu, George (1881–1955, Romania)
Kodály, Zoltán (1882–1967, Hungary)
Webern, Anton (1883–1945, Austria)
Varèse, Edgard (1883–1965, France/USA)
Berg, Alban (1885–1935, Austria)
Martinů, Bohuslav (1890–1959, Czech Republic)
Honegger, Arthur (1892–1955, France/Switzerland)
Milhaud, Darius (1892–1974, France)
Mompou, Federico (1893–1987, Spain)
Gershwin, George (1898–1937, USA)
Poulenc, Francis (1899–1963, France)
Weill, Kurt (1900–1950, Germany)
Rodrigo, Joaquín (1901–1999, Spain)
Walton, William (1902–1983, England)
Tippett, Michael (1905–1998, England)
Messiaen, Olivier (1908–1992, France)
Barber, Samuel (1910–1981, USA)
Cage, John (1912–1992, USA)
Ginastera, Alberto (1916–1983, Argentina)
Bernstein, Leonard (1918–1990, USA)
Piazzolla, Astor (1921–1992, Argentina)
Contemporary/
Late 20th- & Early 21st-Century (10)
Ligeti, György (1923–2006, Hungary/Austria)
Stockhausen, Karlheinz (1928–2007, Germany)
Sondheim, Stephen (b. 1930, USA)
Takemitsu, Tōru (1930–1996, Japan)
Penderecki, Krzysztof (b. 1933, Poland)
Riley, Terry (b. 1935, USA)
Pärt, Arvo (b. 1935, Estonia)
Reich, Steve (b. 1936, USA)
Glass, Philip (b. 1937, USA)
Adams, John (b. 1947, USA)
Since I was being something of a completist I've included all names with at least one popular or iconic work attached to it. So the likes of Allegri, Pachelbel, Lalo, Humperdinck, Dukas, Weill and Riley are on the list, even though they wouldn't crack my all-time top 100 and might indeed rank below some of the near misses unlisted here. This will be an ongoing project till the end of time, so feel free to make suggestions or criticisms as you see fit.
You're welcome.