In brief: at the beginning of theater, choruses produced works for religious occasions and then by the height of drama in the 5th century BC plays were done at civil and religious festivals.

Plays in Ancient Greece evolved over time until they became what we know through the foremost authors: Aeschylus, Euripides, and Sophocles. In the beginning plays were performed by choruses in festivals particularly surrounding the Cult of Dionysius. The early plays were more like passion or mystery plays about religious elements or heroic tales. We don’t know much about the first named playwrights such as Thespis but we do have fragments and whole copies of their works.

Eventually the plays moved from being in the Cult of Dionysius to being performed at festivals all over Greece. For instance in Athens, a festival would always have a dramatic element to it. These festivals were done in competition; three player groups consisting of two, or later three actors, and chorus would compete in four drama sets. The festivals could celebrate a victory in battle, a regular religious event, or any other reason the government would consider, so their frequency is not always standard.

Around the 5th Century is the height of Greek playwriting with the authors Aeschylus, Euripides, and Sophocles writing comedies and tragedies adapting heroic tales and legends. I love the quote describing these plays that “Naturalism was neither achieved nor desired” (Lattimore). These stories would be based on legends and heroic tales. They would never have been tolerated to be about current affairs in Greece or about recent peoples. We do have some plays about Persians and their wars but they were acceptable because they were removed from their society. By the time Euripides comes around, the chorus’ role has been minimized and even disregarded in places.

Many of our current works still follow the formulas for classical tragedy in the tragic hero making their decision and the consequence affecting society until they suffer so it is resolved. Many follow comedic resolution of ending with a marriage or party.

Source: Lattimore, Richard. “Aeschylus I: Oresteia.” 1953. Eds. Allison, Carr, Eastman “Masterpieces of the Drama” fourth ed. Ed. Henderson, Jeffrey "Aristophanses Lysistrata." 1988.

The question of how often plays were produced can vary on a number of things, most notably when in Ancient Greece we are. Theatre evolved from a ritualistic element of festivals to a more standard form of performance over the course of a few centuries, and some kinds of performances were more important than others.

The big deal was the Athenian Festival of Dionysus, also known as the City Dionysia (or just Dionysia), held in the Greek month of Elaphebolion, which translates to around March/April on our calendar. Dionysus was the god of a variety of things, including merriment and the harvest, and Dionysia was a way of praising him for a good harvest. The festival included choral retellings of Greek myths called dithyrambs, honoring the heroes of their believed past, and these choral retellings evolved into being dramatic recreations of these myths—the legend goes that in the 530s BCE Thespis stepped away from his chorus and sang in conversation with them, rather than with them, creating dialogue, while he inhabited the identity of a mythical character, rather than telling the story about that character, though we don't have any sources from the period to confirm this story. As drama became more popular, Dionysia developed the theatric competition, where each day a company performed three tragedies and a satyr play—a short play that basically spoofs tragedy—all written by the same playwright (a different playwright per day), and then judges vote on the winners at the end of the festival. In the mid-5th century, a day was added for comedic plays as well.

But Dionysia wasn't the only time comedy was performed in this era. The smaller Lenaia festival was held in Gamelion (roughly around January), and primarily featured comedy plays. Some of these comedies would later be staged at Dionysia. Comedy was the main focus of Lenaia, though later in the fifth century they added other features, including tragedy. There were also even smaller theatre festivals called Rural Dionysia that were localized to sections of Athens (demes, of which there 130ish) and not designed for the whole city, and were held earlier than the Lenaia. Ian C. Storey and Arlene Allan explain:

One suspects that these productions would be revivials or repeats of earlier plays produced at the major festivals at Athens, to allow those unable to travel to the city to see the plays that they had missed. These were, like the festivals in the city, competitions. The evidence suggests that various deme-theaters preferred one genre or another: Aixone, Rhamnous, and Anagyros seem to have staged only comedy, while Paiania was restricted to tragedy. All three competitions (dithyramb, tragedy, comedy) are known for Eleusis.

So for those of you counting, we are currently at three opportunities to see theatre in the year: once at Lenaia and once at your local deme's Rural Dionysia, and most excitingly once at Dionysia in the spring. A mad man might go from one RD to another (as Plato evidently relates) and up the number, but otherwise, it's just a few times throughout the year. I hope that's enough for you! ^(It's not for me.)

But if you're not in Athens, then are you out of luck? Probably not. Greek writers were able to export their plays across the Mediterranean, so other city-states and non-Greek nations could read and perform these texts. Chris Dearden notes, for example, that the plays of Epicharmus (a late 6th/early 5th century dramatist) were performed in Syracuse throughout the fifth century, and works by the Athenian playwright Euripides were being performed in Sicily toward the end of the century. It seems likely that local troupes performed these plays, rather than exporting actors with the plays, though it does seem that Athenian actors did occasionally travel to and perform at other city-states. They also wound up writing some of their own plays. Plato tells us that in Sicily and Southern Italy, theatre was used in competition similar to in Athens.

As I understand it, theatre troupes at some point developed in Athens, such that performances were not limited to just the festivals. Whether that was a thing in the fifth century, or not until the fourth, I'm not quite sure. Gradually, though, theatre grew more common, and new theater buildings were propping up all across Athens, and opportunities to see theatre increased further and further. Tragic playwrights stopped relying on drawing from history and legends, and started telling more original stories. And in the middle of the fourth century, the Dionysia added a performance dedicated to staging older plays, making it easier for people to see the great classic plays by Euripides, Aeschylus, and Sophocles, who dominated the fifth century competitions.

My personal study has primarily focused on how the Dionysia worked in the fifth century, so there are a lot of elements about non-Athenian theatre, lower level festivals and travelling troupes, and the Hellenistic- and more so Roman-period theatre that are certainly beyond me. But as I've attempted to overview—synthesizing some information that's old to me and some that's more fresh—as theatre grew more popular, so too did how often plays were performed and people were able to see theatre.

^(Edit: some proofreading)

Further Reading

Storey, Ian C. and Arlene Allan. “Aspects of Ancient Greek Drama.” A Guide to Ancient Greek Drama, Wiley Blackwell, Oxford, 2015, www.google.com/books/edition/A_Guide_to_Ancient_Greek_Drama/z2t1cIVk-A8C?hl=en&gbpv=1&pg=PA15&lpg=PA15. Accessed 14 Jan. 2021.

Dearden, Chris. “Plays for Export.” Phoenix, vol. 53, no. 3/4, 1999, p. 222., doi:10.2307/1088985.

Green, J. R. Theatre in Ancient Greek Society. Routledge, 1994.

Kotlińska-Toma Agnieszka. Hellenistic Tragedy: Texts, Translations and a Critical Survey. Bloomsbury, 2016.