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It seems likely that Círdan is the oldest living Elf at the end of the Third Age, see here.
Can we estimate how old he is?
I don't think we can get an exact figure, since we don't know when exactly he was born. So I'm interested in an approximation/a range of possibilities.
Using this timeline from Annals of Aman (HoME 10) and its conversion of 1 Year of the Trees = 9.582 Years of the Sun.
Círdan was born at some point between 1050 YT (when the Elves awoke) and 1115 YT (the Second Sundering). An essay in HoME 12 notes that Círdan was the foremost in the craft of boat-making while still East of the Misty Mountains.
From the time the Sun rose (1500 YT), the three Ages were 590, 3441, and 3021 years long, respectively.
So his total age at the end of the Third Age was between 10741 and 11364 years old. And indeed nobody (Elf, Man, Dwarf, Ent, or Orc) could be older than that upper bound.
You could use a different YT conversion from other parts of Tolkien's work, but then the whole timeline probably needs changing anyway.
Tolkien worked in his annals off and on for decades.
That included changing dates for various events. Thus it is possible that the number of years between when Cirdan was first mentioned and the first Year of the Sun may vary in different versions.
Tolkien also tried several different ratios between the Years of the Trees (YT) used in those annals and Years of the Sun (YS).
If I remember correctly, one version made one YT equal to nine and a fraction YS, another made a YT equal 10 YS, and another version made a YT equal one loa, which in turn equaled 144 YS.
And i am not an expert on that part of the legendarium, and I don't know if Tolkien made a final decision which of those ratios - or maybe some other ratio - was the correct one.
So I don't know how many thousands, or tens of thousands, or hundreds of thousands, of time spans equal to Years of the Sun Cirdan lived before the first year of the Sun.
I don't know exactly how long the first Age of the Sun lasted, about 600 hundred YS in the latest version, if I remember correctly. The Second Age lasted for 3,441 years and the Third Age lasted for 3,021 years before Cirdan greeted Gandalf, Frodo, Bilbo, Elrond, Galadriel, etc. at the Gray haven, which was thus 6,462 years after the end of the First Age and about 7,062 years after the first rising of the Sun.
Added 09-09-2021.
OrangeDog's answer says that according to the Annals of Aman the first Elves awoke in YT 1050, Cirdan must have been born by YT 1115, and the Sun rose in YT 1500, and that the First Age of the Sun lasted for 590 years of the Sun.
So that makes Cirdan aged 7,052 Years of the Sun plus 385 to 450 Years of the Trees in Third Age 3021.
OrangeDog uses a ratio of 9.582 Years of the Sun (YS)per each Year of the Trees (YT), and so he calculates that Cirdan was between 10,741 and 11,364 YS old in Third Age 3021.
Using the ratio of 10 YS to 1 YT, Cirdan would be 3,850 to 4,500 years old at the first rising of the Sun, and would have a total age of 10,902 to 11,552 years in Third age 3021.
Using the ratio of 144 YS to 1 YT, Cirdan would be 55,440 to 64,800 years old at the first rising of the Sun, and would have a total age of 62,792 to 71,852 years in Third age 3021.
There might be other ratios mentioned in various writings, and I expect that there were other dates in the various versions of the annals. So there should be several other age ranges of Cirdan in Third Age 3021 that could be calculated.
To answer according to Tolkien's later expanded chronology of the first age:
The closest I can find to a statement about Círdan's birth is this statement from c.1972-3 that he was "akin" to Olwë and Elwë.
Círdan was a Telerin Elf, one of the highest of those who were not transported to Valinor but became known as the Sindar, the Grey-elves; he was akin to Olwë, one of the two kings of the Teleri, and lord of those who departed over the Great Sea. He was thus also akin to Elwë, Olwë's elder brother, acknowledged as high-king of all the Teleri in Beleriand, even after he withdrew to the guarded realm of Doriath.
The Peoples of Middle-earth - "Last Writings: Círdan"
This is similar to something Tolkien said previously in 1959-60:
Elwe himself had indeed long and beautiful hair of silver hue, but this does not seem to have been a common feature of the Sindar, though it was found among them occasionally, especially in the nearer or remoter kin of Elwe (as in the case of Círdan).
The War of the Jewels - "Quendi and Eldar"
In a footnote on the above text, Christopher Tolkien says that he has "not found any statement of the nature of the kinship", so it's probably not a lot to go on, but lacking anything else it is what we will go on.
As it happens, we do have exact dates of birth for both Olwë and Elwë, given around the same time as when Tolkien first said Círdan and Elwë were kin. At this point Tolkien had decided that the First Age would start counting from the awakening of the elves. (The Sun was around from the beginning and so no longer made sense as a starting point.)
“DB” [Days of Bliss] still goes on, but Quendi start reckoning of First Age with Awaking.
First Age must last somewhat longer than SA (= 3,441). Still be more regularly “duodecimal” (as mythological) up to Death of Trees and after! Say, 4,056 years.
The Nature of Middle-earth - "Key Dates"
Tolkien says that Elwë and Olwë were born in years 2126 and 2185 of the First Age, thus making them 1,930 and 1,871 years old at the end of the First Age, had they both survived.
The Embassy must be later, not more than 20 years before the March, allowing 10 years of absence and 10 years of preparation: say 2212. Let Ingwë be born in 2072; Finwë in 2120 (48 years later); and Elwë in 2126. Then at the Embassy Ingwë would be 140, Finwë 92, and Elwë 86. At the beginning of the March Ingwë would be 160, Finwë 112, and Elwë 106. ... Finwë and Elwë were friends, very adventurous. Olwë, born 2185, was 27 at the Embassy, 47 at the March (Elmo was born on the March).
The Second and Third Ages lasted for a combined 6,462 years, so if Círdan were born at the same time as his kinsmen Elwë and Olwë, he would have been around 8,300 years old at the end of the Third Age.
In terms of hard upper and lower bounds, Círdan would have to have been born after the awakening of the elves (i.e. younger than 10,518), and before Elwë was lost at the end of the Great March, as he was already a leader then.
It was during the long waiting of the Teleri for the return of the floating isle, upon which the Vanyar and Noldor had been transported over the Great Sea, that Círdan had turned his thoughts and skill to the making of ships, for he and all the other Teleri became impatient. Nonetheless it is said that for love of his kin and allegiance Círdan was the leader of those who sought longest for Elwë when he was lost and did not come to the shores to depart from Middle-earth. Thus he forfeited the fulfilment of his greatest desire: to see the Blessed Realm and find again there Olwë and his own nearest kin. Alas, he did not reach the shores until nearly all the Teleri of Olwë's following had departed.
The Peoples of Middle-earth - "Last Writings: Círdan"
I can't find a great date for Elwë being lost that seems consistent with the other material in this answer (Tolkien was changing his mind on this stuff often and the various time schemes usually do not reach that point.), but it looks like it may have been 4 Valian Years after the Great March started (FA 2232), which would be FA 2808, and thus set a lower bound on him being older than ~7,700 years at the end of the Third Age.
