How many dragons ever existed in Westeros?

Question

There are multiple questions relevant to each other:

1. How many dragons ever existed in Westeros?
2. How do they end up?
3. When did the number reach its peak?

Answer 1

Number of Dragons in Westeros

In Westeros Dragons first came with Targaryen invaders. There were three of those dragons:

1. Balerion the Black Dread
2. Vhagar
3. Meraxes

In reign of King Viserys I, twenty dragons were alive. Since records aren't kept for all hatchlings, we only have records of dragons which grew up a bit. Many hatchlings died in infancy and have no record. Many hatchlings were eaten by other dragons. So following would account only for known dragons:

╔═════════════╦═══════════════════╗
║ Monarch     ║ Number of Dragons ║
╠═════════════╬═══════════════════╣
║ Aegon I     ║ 3                 ║
╠═════════════╬═══════════════════╣
║ Jaehaerys I ║ 8                 ║
╠═════════════╬═══════════════════╣
║ Viserys I   ║ 20                ║
╠═════════════╬═══════════════════╣
║ Aegon III   ║ 4                 ║
╚═════════════╩═══════════════════╝

By taking only the distinct dragons from sets of these monarchs we get the final estimate of 25.

So it is safe to say that there were 25 known dragons in Westeros.

The complete list of known Dragons, their riders and their fate would be:

1. Balerion the Black Dread - Ridden by Kings Aegon I, Maegor I and Viserys I along with Princess Aerea Targaryen (Grand daughter of King Aenys I). Died of Natural causes during reign of King Jaehaerys I.
2. Vhagar - Ridden by Queen Visenya Targaryen, Prince Baelon Targaryen (son of King Jaehaerys I), Laena Velaryon and Prince Aemond Targaryen (son of King Viserys I). Killed in combat against Caraxes
3. Meraxes - Ridden by Queen Rhaenys Targaryen. Killed in combat against Armies of Dorne
4. Quicksilver - Ridden by King Aenys I and Prince Aegon Targaryen (Aenys I's son). Killed in combat against Balerion
5. Vermithor the Bronze Fury - Ridden by King Jaehaerys I and Hugh the Hammer. Killed in fight against Seasmoke and Tessarion
6. Silverwing - Ridden by Queen Alysanne the Good and Ulf the Sot. Went mad with grief on Vermithor's death and fled to the wild
7. Dreamfyre - Ridden by Princess Rhaena Targaryen (Aenys I's daughter) and Queen Helaena Targaryen. Killed by rabble during Storming of Dragonpit
8. Meleys the Red Queen - Ridden by Princess Alyssa Targaryen (daughter of King Jaehaerys I) and her niece Princess Rhaenys Targaryen, Queen who never was . Killed in combat against Vhagar and Sunfyre
9. Caraxes the Blood Wyrm - Ridden by Prince Aemon Targaryen (son of King Jaehaerys I) and Prince Daemon Targaryen (Grand son of King Jaehaerys I). Killed in fight against Vhagar
10. Syrax - Ridden by Princess Rhaenyra Targaryen (Daughter to King Viserys I). Killed by rabble during Storming of Dragonpit
11. Seasmoke - Ridden by Sers Laenor Velaryon and Addam Velaryon. Killed in fight against Vermithor and Tessarion
12. Sunfyre the Golden - Ridden by King Aegon II. Paralyzed with wounds taken in fight against Meleys. Recovered and took some wounds in fight against Grey Ghost. Died of fatal wounds taken in fight against Moon Dancer
13. Tessarion the Blue Queen - Ridden by Prince Daeron Targaryen (son of King Viserys I). Took fatal wounds in fight against Vermithor and Seasmoke. Killed on orders of Lord Benjicot Blackwood the next day to put an end to her suffering.
14. Vermax - Ridden by Prince Jacaerys Velaryon (Grand son of King Viserys I). Killed in fight against Fleet of Kingdom of Three daughters
15. Arrax - Ridden by Prince Lucerys Velaryon (Grand son of King Viserys I). Killed in fight against Vhagar
16. Tyraxes - Ridden by Prince Joffrey Velaryon (Grand son of King Viserys I). Killed by rabble during Storming of Dragonpit
17. Moon Dancer - Ridden by Princess Baela Targaryen (daughter of Prince Daemon). Killed in fight against Sunfyre
18. Stormcloud - Ridden by King Aegon III. Died of wounds taken during escape from attacking fleet of Kingdom of Three daughters
19. Morghul - Unridden. Bounded to Queen Jaehaera Targaryen. Killed by rabble during Storming of Dragonpit
20. Shrykos - Unridden. Bounded to Prince Jaehaerys Targaryen (Aegon II's son). Killed by rabble during Storming of Dragonpit
21. Grey Ghost - Unridden. Killed by Sunfyre
22. Sheep Stealer - Ridden by Nettles. Fled Westeros, never to be seen again - Rumoured to have made mountains of the Moon its lair
23. Cannibal - Unridden. Disappeared mysteriously
24. The Last Dragon - Unridden. Died of natural causes
25. Morning - Ridden by Princess Rhaena Targaryen (Daughter of Prince Daemon). Appears to have died of natural causes

After King Viserys I's death, a civil war broke out between his daughter Rhaenyra Targaryen and his son Aegon II which is known as the Dance of the Dragons. Almost all of the above mentioned fighting occurred during that civil war (With Exception of Battle between Meraxes and Dornish Armies and Battle between Balerion and Quicksilver). Most dragons died as a result.

To see which dragons survived the Dance of the Dragons, see my previous answer here. To see complete kill-score of the dragons and their riders in the Dance, see my previous answer here.

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Maximum number of Dragons in Westeros at one time

Dragons reached their maximum number under reign of King Viserys I (103-129 AC).

When the King Viserys went to Dragonstone for wedding of his daughter Rhaenyra Targaryen, it is said the Island became new Valyria because of the high number of Dragons.

There were twenty dragons alive when King Viserys I died. Those included:

1. Vhagar
2. Vermithor
3. Silverwing
4. Dreamfyre
5. Meleys
6. Caraxes
7. Syrax
8. Seasmoke
9. Sunfyre
10. Tessarion
11. Vermax
12. Arrax
13. Tyraxes
14. Moon Dancer
15. Stormcloud
16. Morghul
17. Shyrkos
18. Grey Ghost
19. Sheep Stealer
20. Cannibal

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Second highest number of Dragons in Westeros

The second highest number would be during reign of King Jaehaerys I (48-103 AC), in which at least 8 Dragons were alive. It is said during the great tourney of King's Landing, there were so many dragons that King's Landing appeared to be new Valyria. Those 8 dragons might have been:

1. Balerion the Black Dread (Confirmed)
2. Vhagar (Confirmed)
3. Vermithor (Confirmed)
4. Silverwing (Confirmed)
5. Dreamfyre (Confirmed)
6. Meleys (Confirmed)
7. Caraxes (Confirmed)
8. Syrax (Probable)

NOTE: There are legends about a Dragon Urrax who may have lived in time of the first men but we will not enter him to our list unless GRRM confirms existence of Urrax and of the Knight Ser Serwyn of Mirror Shield who killed it. It appears to be a fabricated story because in times of Ser Serwyn, there were no Knights in Westeros. Knights came with Andals who did not exist in Westeros at that time. Also, The legends say he may have been a Kingsguard but that is also false because Order of Kingsguard was established after the Conquest by Queen Visenya. We will discuss only confirmed dragons, which are Targaryen dragons.

Answer 2

20 Dragons were the most to be alive in Westeros at any one time. Most were lost during The Dance of the Dragons (129 to 131 AC) and the last dragon died in 153 AC, until the re-birth of Daenerys' three.

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Dragons: A History

Dragons were discovered some five thousand years ago by the Valyrians in the Fourteen Flames, a ring of volcanoes on the Valyrian peninsula.

After the Doom of Valyria, the only dragons known to have survived were five dragons on Dragonstone. The dragons belonged to House Targaryen, who had brought them from Valyria when they went into exile. Four of these dragons eventually died on Dragonstone, leaving only Balerion. However, the other dragons left eggs behind, and from these hatched Vhagar and Meraxes.

House Targaryen used Balerion, Vhagar, and Meraxes in their Wars of Conquest against the Seven Kingdoms.

Many dragons were hatched after Aegon's Conquest. By 129 AC, on the eve of civil war between Rhaenyra and Aegon II, there were twenty living dragons.

The majority of these died during the civil war called the Dance of the Dragons, which began in 129 AC and would last until mid 131 AC. By the end of the Dance in 131 AC, only four dragons remained alive.

The last dragon was a stunted, sick and misshapen thing, and she died young, in 153 AC. She had been a green female, small with withered wings. She laid a clutch of five eggs, which never hatched.

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So, we can extrapolate from this that before the Doom of Valyria that dragons probably numbers in the hundreds or even possibly thousands. Records from that time period are notoriously incomplete as most of them were destroyed or otherwise lost.

In Westeros the maximum number of dragons at one time was 20. Although, it only took three to conquer it (hmmm... foreshadowing anyone...).