There are multiple questions relevant to each other:
- How many dragons ever existed in Westeros?
- How do they end up?
- When did the number reach its peak?
In Westeros Dragons first came with Targaryen invaders. There were three of those dragons:
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:
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.
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:
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:
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.
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.
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.
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...).