I found 15 lists of the best plays (best theater, best drama) of all time on the Internet and in books and compiled them all into one meta-list.  The results are below: every play that is on at least two of the original source lists, organized chronologically. To see a smaller list of the top plays organized by rank (that is, with the plays on the most lists at the top), go here.

A few notes:
(1) As is so often the case, the lists I am able to find online have a definite bias in favor of Western Civilization generally, and British and American culture specifically. Nevertheless, the lists I found contained a significant number of works that were originally written in a language other than English (including the top vote-getter: Waiting for Godot – Irish-born playwright Samuel Beckett wrote the original in French and the first production of the play was in French; he later translated it to his native English).
(2) The date is usually the date of the premiere production, unless the play was written and published before the first production, in which case I use the earlier date.
(3) Nearly all the lists I found excluded musicals, so I did not include musicals on the list, with the exception of the Brecht/Weill masterpiece, The Threepenny Opera.
(4) For those of you who thought this was a list of the best sports plays of all time, I apologize for the inconvenience.

500 BCE-1 BCE
The Persians (Ancient Greece, 472 BCE). By Aeschylus (on 2 lists)
The Oresteia (Ancient Greece, 458 BCE). By Aeschylus (on 6 lists)
Antigone (Ancient Greece, c. 441 BCE). By Sophocles (on 5 lists)
Medea (Ancient Greece, 431 BCE). By Euripides (on 9 lists)
Oedipus the King (Ancient Greece, c. 429 BCE). By Sophocles (on 12 lists)
The Trojan Women (Ancient Greece, 415 BCE). By Euripides (on 3 lists)
Electra (Ancient Greece, 413 BCE). By Sophocles (on 2 lists)
Lysistrata (Ancient Greece, 411 BCE). By Aristophanes (on 5 lists)
The Bacchae (Ancient Greece, 405 BCE). By Euripides (on 5 lists)
The Frogs (Ancient Greece, 405 BCE). By Aristophanes (on 2 lists)
Menaechmi (The Brothers Menaechmus) (Ancient Rome, c. 205-184 BCE). By Plautus (on 2 lists)

Joel McDermott, Danny Sapani, Jude Pearce and Helen McCrory in a 2014 production of Medea at the National Theatre in London. Photo by Richard Hubert Smith.

1 CE-1499
The Recognition of Shakuntala (India, c. 100 BCE-400 CE). By Kalidasa (on 3 lists)
Izutsu (The Well Curb; The Well Cradle) (Japan, c. 1420-1440). By Zeami Motokiyo (on 2 lists)
Everyman (England, c. 1485-1499). By Anonymous (on 2 lists)

A scene from a 2005 production of The Recognition of Shakuntala in Texas, US.

1500-1599
The Taming of the Shrew (England, c. 1590-1591). By William Shakespeare (on 2 lists)
Richard III (England, c. 1591-1592). By William Shakespeare (on 4 lists)
Titus Andronicus (England, c. 1591-1592). By William Shakespeare (on 2 lists)
Doctor Faustus (England, c. 1592). By Christopher Marlowe (on 7 lists)
Edward II (England, 1593-1594). By Christopher Marlowe (on 3 lists)
Romeo and Juliet (England, c. 1591-1595). By William Shakespeare (on 5 lists)
Love’s Labor Lost (England, 1594-1595). By William Shakespeare (on 2 lists)
Richard II (England, 1595). By William Shakespeare (on 2 lists)
A Midsummer Night’s Dream (England, c. 1595-1596). By William Shakespeare (on 7 lists)
The Merchant of Venice (England, c. 1596-1597). By William Shakespeare (on 2 lists)
Henry IV, Part One (England, c. 1597). By William Shakespeare (on 4 lists)
Henry IV, Part Two (England, c. 1597-1598). By William Shakespeare (on 2 lists)
The Peony Pavilion (China, 1598). By Tang Xiansu (on 2 lists)
Much Ado About Nothing (England, 1598-1599). By William Shakespeare (on 2 lists)
Henry V (England, c. 1599). By William Shakespeare (on 3 lists)

Arthur Darville and Paul Hilton in a 2011 production of Doctor Faustus at Shakespeare’s Globe Theater in London.

1600-1699
As You Like It (England, c. 1599-1600). By William Shakespeare (on 2 lists)
Hamlet (England, c. 1599-1602). By William Shakespeare (on 12 lists)
Twelfth Night (England, c. 1601-1602). By William Shakespeare (on 7 lists)
Othello (England, c. 1603). By William Shakespeare (on 6 lists)
Volpone (England, c. 1605-1606). By Ben Jonson (on 4 lists)
Macbeth (England, c. 1606). By William Shakespeare (on 7 lists)
King Lear (England, 1606). By William Shakespeare (on 7 lists)
Antony & Cleopatra (England, c. 1606-1607). By William Shakespeare (on 4 lists)
Coriolanus (England, c. 1605-1608). By William Shakespeare (on 4 lists)
The Winter’s Tale (England, 1609-1610). By William Shakespeare (on 2 lists)
The Alchemist (England, 1610). By Ben Jonson (on 3 lists)
The Tempest (England, 1610-1611). By William Shakespeare (on 4 lists)
The White Devil (England, 1612). By John Webster (on 2 lists)
Fuenteovejuna (Spain, c. 1612-1614). By Lope de Vega (on 3 lists)
Life Is a Dream (Spain, 1635). By Pedro Calderón de la Barca (on 4 lists)
Le Cid (France, 1636). By Pierre Corneille (on 3 lists)
Tartuffe (France, 1664). By Molière (on 8 lists)
The Misanthrope (France, 1666). By Molière (on 4 lists)
Andromache (France, 1667). By Jean Racine (on 2 lists)
The Miser (France, 1668). By Molière (on 2 lists)
The Country Wife (England, 1675). By William Wycherley (on 2 lists)
The Rover (England, 1677). By Aphra Behn (on 3 lists)
Phèdre (France, 1677). By Jean Racine (on 3 lists)

Anastasia Hille and Benedict Cumberbatch in a 2015 production of Hamlet at the Barbican Theater in London. Johan Persson photo.

1700-1799
The Love Suicides at Sonezaki (Japan, 1703). By Chikamatsu Monzaemon (on 2 lists)
The Servant of Two Masters (One Man, Two Guvnors) (Italy, 1746). By Carlo Goldoni (on 2 lists)
She Stoops to Conquer (Ireland, 1773). By Oliver Goldsmith  (on 5 lists)
The School for Scandal (England, 1777). By Richard Brinsley Sheridan (on 5 lists)
The Marriage of Figaro (France, 1778). By Pierre Beaumarchais (on 2 lists)
Nathan the Wise (Germany, 1779). By Gotthold Ephraim Lessing  (on 2 lists)

A scene from a 2015 production of She Stoops to Conquer at the Avon Theatre in Stratford, Ontario, Canada. Photo by David Hou.

1800-1899
Mary Stuart (Germany, 1800). By Friedrich Schiller (on 2 lists)
The Prince of Homburg (Germany, 1809-1810 (written), 1821 (performed)). By Heinrich von Kleist  (on 2 lists)
Faust (Germany, 1808 (Pt. 1), 1831 (Pt. 2)). By Johann Wolfgang von Goethe (on 3 lists)
The Government Inspector (The Inspector General) (Russia, 1836). By Nikolai Gogol (on 5 lists)
Woyzeck (Germany, 1836-1837). By Georg Büchner (on 6 lists)
A Doll’s House (Norway, 1879). By Henrik Ibsen (on 10 lists)
Ghosts (Norway, 1882). By Henrik Ibsen (on 2 lists)
An Enemy of the People (Norway, 1883). By Henrik Ibsen (on 2 lists)
The Wild Duck (Norway, 1884). By Henrik Ibsen (on 4 lists)
The Father (Sweden, 1887). By August Strindberg (on 2 lists)
Miss Julie (Sweden, 1888). By August Strindberg (on 7 lists)
Hedda Gabler (Norway, 1891). By Henrik Ibsen (on 7 lists)
The Weavers (Germany, 1892). By Gerhart Hauptmann  (on 2 lists)
Arms and the Man (UK, 1894). By George Bernard Shaw (on 3 lists)
The Importance of Being Earnest (Ireland, 1895). By Oscar Wilde (on 12 lists)
The Seagull (Russia, 1895). By Anton Chekhov (on 4 lists)
Ubu Roi (King Ubu) (France, 1896). By Alfred Jarry  (on 3 lists)
Uncle Vanya (Russia, 1897). By Anton Chekhov (on 8 lists)
Cyrano de Bergerac (France, 1897). By Edmond Rostand (on 3 lists)

Allan Aynesworth, Evelyn Millard, Irene Vanbrugh and George Alexander in the 1895 London premiere production of The Importance of Being Earnest.

1900-1949
La Ronde (Austria, 1900 (published), 1920 (performed)). By Arthur Schnitzler (on 2 lists)
Three Sisters (Russia, 1901). By Anton Chekhov (on 6 lists)
Man and Superman (UK, 1901-1902). By George Bernard Shaw (on 4 lists)
The Lower Depths (Russia, 1902). By Maxim Gorky (on 4 lists)
Peter Pan (UK, 1904). By J. M. Barrie (on 2 lists)
The Cherry Orchard (Russia, 1904). By Anton Chekhov (on 7 lists)
Major Barbara (UK, 1905). By George Bernard Shaw (on 3 lists)
A Flea in Her Ear (France, 1907). By Georges Feydeau (on 2 lists)
The Playboy of the Western World (Ireland, 1907). By J.M. Synge (on 7 lists)
The Ghost Sonata (Sweden, 1908). By August Strindberg (on 2 lists)
Pygmalion (UK, 1913). By George Bernard Shaw (on 7 lists)
The Post Office (India, 1913). By Rabindranath Tagore (on 2 lists)
Heartbreak House (UK, 1919). By George Bernard Shaw (on 3 lists)
The Dybbuk (Poland/Russia/Belarus, 1920). By S. Ansky (on 2 lists)
Six Characters in Search of an Author (Italy, 1921). By Luigi Pirandello (on 6 lists)
Henry IV (Italy, 1922). By Luigi Pirandello (on 3 lists)
Saint Joan (UK, 1923). By George Bernard Shaw (on 4 lists)
Juno and the Paycock (Ireland, 1924). By Seán O’Casey (on 3 lists)
The Vortex (UK, 1924). By Noël Coward (on 2 lists)
The Plough and the Stars (Ireland, 1926). By Seán O’Casey (on 2 lists)
Machinal (US, 1928). By Sophie Treadwell (on 4 lists)
Journey’s End (UK, 1928). By R.C. Sherriff (on 2 lists)
The Threepenny Opera (Germany, 1928). By Bertolt Brecht & Kurt Weill (on 4 lists)
The Front Page (US, 1928). By Ben Hecht and Charles MacArthur (on 2 lists)
Mourning Becomes Electra (US, 1931). By Eugene O’Neill (on 2 lists)
Private Lives (UK, 1930). By Noël Coward (on 4 lists)
Design for Living (UK, 1930). By Noël Coward (on 2 lists)
Blood Wedding (Spain, 1933). By Federico García Lorca (on 4 lists)
The Children’s Hour (US, 1934). By Lillian Hellman (on 2 lists)
Awake and Sing! (US, 1935). By Clifford Odets (on 4 lists)
Murder in the Cathedral (US/UK, 1935). By T.S. Eliot (on 2 lists)
The Women (US, 1936). By Claire Booth Luce (on 2 lists)
The House of Bernarda Alba (Spain, 1936). By Federico Garcia Lorca (on 2 lists)
Golden Boy (US, 1937). By Clifford Odets (on 2 lists)
Electra (France, 1937). By Jean Giraudoux (on 2 lists)
Our Town (US, 1938). By Thornton Wilder (on 9 lists)
Mother Courage and Her Children (Germany, 1939). By Bertolt Brecht (on 7 lists)
The Little Foxes (US, 1939). By Lillian Hellman (on 4 lists)
The Time of Your Life (US, 1939). By William Saroyan (on 2 lists)
The Philadelphia Story (US, 1939). By Philip Barry (on 2 lists)
Blithe Spirit (UK, 1941). By Noël Coward (on 3 lists)
The Good Person of Szechwan (Germany, 1941). By Bertolt Brecht (on 4 lists)
Present Laughter (UK, 1942). By Noel Coward (on 3 lists)
The Life of Galileo (Germany, 1943). By Bertolt Brecht (on 2 lists)
No Exit (France, 1944). By Jean-Paul Sartre (on 6 lists)
The Glass Menagerie (US, 1944). By Tennessee Williams (on 9 lists)
The Iceman Cometh (US, 1946). By Eugene O’Neill (on 6 lists)
A Streetcar Named Desire (US, 1947). By Tennessee Williams (on 11 lists)
All My Sons (US, 1947). By Arthur Miller (on 3 lists)
The Caucasian Chalk Circle (Germany/US, 1948). By Bertolt Brecht (on 2 lists)
Mr. Puntila and His Man Matti (Germany/US, 1948). By Bertolt Brecht (on 2 lists)
Death of a Salesman (US, 1949). By Arthur Miller (on 12 lists)

Marlon Brando, Kim Hunter and Jessica Tandy in a scene from the original 1947 production of A Streetcar Named Desire at the Ethel Barrymore Theater in New York.

1950-1999
The Bald Soprano (Romania/France, 1950). By Eugène Ionesco (on 7 lists)
The Chairs (Romania/France, 1952). By Eugène Ionesco  (on 2 lists)
The Deep Blue Sea (UK, 1952). By Terence Rattigan (on 2 lists)
Waiting for Godot (Ireland/France,1953) By Samuel Beckett (on 14 lists)
The Crucible (US, 1953). By Arthur Miller (on 7 lists)
Picnic (US, 1953). By William Inge (on 3 lists)
Twelve Angry Men (US, 1954 (teleplay), 1964 (stage)). By Reginald Rose (on 2 lists)
Cat on a Hot Tin Roof (US, 1955). By Tennessee Williams (on 4 lists)
A View from the Bridge (US, 1955). By Arthur Miller (on 4 lists)
Long Day’s Journey into Night (US, 1956). By Eugene O’Neill (on 11 lists)
The Visit (Switzerland, 1956). By Friedrich Dürrenmatt  (on 4 lists)
Look Back in Anger (UK, 1956). By John Osborne (on 6 lists)
The Birthday Party (UK, 1957). By Harold Pinter (on 5 lists)
Endgame (Ireland/France, 1957). By Samuel Beckett (on 3 lists)
The Balcony (France, 1957). By Jean Genet (on 3 lists)
The Entertainer (UK, 1957). By John Osborne (on 2 lists)
The Dark at the Top of the Stairs (US, 1957). By William Inge (on 2 lists)
The Zoo Story (US, 1958). By Edward Albee (on 2 lists)
Serjeant Musgrave’s Dance (UK, 1959). By John Arden  (on 2 lists)
Rhinoceros (Romania/France, 1959). By  Eugène Ionesco (on 4 lists)
A Raisin in the Sun (US, 1959). By Lorraine Hansberry (on 8 lists)
The Caretaker (UK, 1960). By Harold Pinter (on 3 lists)
Blood Knot (South Africa, 1961). By Athol Fugard (on 2 lists)
Who’s Afraid of Virginia Woolf? (US, 1962). By Edward Albee (on 8 lists)
The Homecoming (UK, 1964). By Harold Pinter (on 7 lists)
Marat/Sade (Germany, 1964) By Peter Weiss (on 4 lists)
Dutchman (US, 1964). By LeRoi Jones (Amiri Baraka) (on 2 lists)
The Odd Couple (US, 1965). By Neil Simon (on 2 lists)
Rosencrantz and Guildenstern Are Dead (UK, 1966). By Tom Stoppard (on 5 lists)
A Day in the Death of Joe Egg (UK, 1967). By Peter Nichols (on 2 lists)
What the Butler Saw (UK, 1969). By Joe Orton (on 3 lists)
Accidental Death of an Anarchist (Italy, 1970). By Dario Fo (on 4 lists)
The House of Blue Leaves (US, 1971). By John Guare (on 2 lists)
The Norman Conquests (trilogy) (UK, 1973). By Alan Ayckbourn (on 2 lists)
Equus (UK, 1973). By Peter Shaffer (on 4 lists)
Travesties (UK, 1974). By Tom Stoppard (on 3 lists)
American Buffalo (US, 1975). By David Mamet (on 4 lists)
Death and the King’s Horseman (Nigeria, 1975). By Wole Soyinka (on 2 lists)
Uncommon Women and Others (US, 1977). By Wendy Wasserstein (on 3 lists)
Betrayal (UK, 1978). By Harold Pinter (on 4 lists)
Buried Child (US, 1978). By Sam Shepard (on 2 lists)
Amadeus (UK, 1979). By Peter Shaffer (on 2 lists)
Cloud 9 (UK, 1979). By Caryl Churchill (on 4 lists)
True West (US, 1980). By Sam Shepard (on 3 lists)
’night, Mother (US, 1982). By Marsha Norman (on 2 lists)
The Real Thing (UK, 1982). By Tom Stoppard (on 3 lists)
“Master Harold”… and the Boys (South Africa, 1982). By Athol Fugard (on 7 lists)
Top Girls (UK, 1982). By Caryl Churchill  (on 4 lists)
Noises Off (UK, 1982). By Michael Frayn (on 4 lists)
Glengarry Glen Ross (US, 1984). By David Mamet (on 6 lists)
Fool for Love (US, 1984). By Sam Shepard (on 2 lists)
The Normal Heart (1985). By Larry Kramer (on 3 lists)
Fences (US, 1985). By August Wilson (on 5 lists)
The Heidi Chronicles (US, 1988). By Wendy Wasserstein (on 2 lists)
M Butterfly (US, 1988). By David Henry Hwang (on 4 lists)
Six Degrees of Separation (US, 1990). By John Guare (on 3 lists)
Dancing at Lughnasa (Ireland, 1990). By Brian Friel  (on 3 lists)
The Other Shore (China, 1990). By Gao Xingjian (on 2 lists)
Angels in America (US, 1993). By Tony Kushner (on 8 lists)
Arcadia (UK, 1993). By Tom Stoppard (on 5 lists)
The Weir (Ireland/UK, 1997). By Conor McPherson (on 2 lists)

Ruby Dee, Claudia McNeil, Glynn Turman, Sidney Poitier and John Fiedler in the original 1959 production of A Raisin in the Sun in New York.

2000-Present
Proof (US, 2000). By David Auburn (on 2 lists)
The Lieutenant of Inishmore (Ireland/UK, 2001). By Martin McDonagh (on 2 lists)
Topdog/Underdog (US, 2001). By Suzan-Lori Parks (on 5 lists)
The Goat, or Who Is Sylvia? (US, 2002). By Edward Albee (on 2 lists)
The Pillowman (Ireland, 2003). By Martin McDonagh (on 2 lists)
The History Boys (UK, 2004). By Alan Bennett  (on 3 lists)
The Clean House (US, 2004). By Sarah Ruhl (on 2 lists)
Doubt (US, 2004). By John Patrick Shanley (on 3 lists)
God of Carnage (France, 2006). By Yasmina Reza (on 2 lists)
August: Osage County (US, 2007). By Tracy Letts (on 3 lists)
War Horse (UK, 2007). By Nick Stafford (on 2 lists)
Ruined (US, 2009). By Lynn Nottage (on 3 lists)
Red (US, 2009). By John Logan (on 3 lists)
Jerusalem (UK, 2009). By Jez Butterworth (on 3 lists)
Enron (UK, 2009). By Lucy Prebble (on 2 lists)
Clybourne Park (US, 2010). By Bruce Norris (on 3 lists)
The Curious Incident of the Dog in the Night-Time (UK, 2012). By Simon Stephens (on 2 lists)

Cherry Jones and Brian F. O’Byrne in the original 2004 production of Doubt at the Walter Kerr Theater in New York.  Photo by Sara Krulwich/New York Times.