How many times has someone taken birth into samsara?

Where did Buddha says that "a beginning point is not evident"? If beginning point is not evident then the world has been manifesting since eternity.

I'm not sure that "not evident" is meant to imply "infinite" or "since eternity".

Even "5 minutes ago" isn't very evident, at least to me, "500 years ago" is even less evident, and "London and New York" are scarcely evident from here, etc.

It may be better to consider what's akalika -- timeless, eternal, ever-present.

If you're in a court of law and someone tells you that a fact is "not evident", I think that means that there's "no evidence": which I guess implies that it's a question that you cannot answer, perhaps therefore a wrong or useless question (unless you're adding some additional debatable context-specific assumptions, like "innocent until proven guilty" therefore "no evidence implies innocence"; or "everything that ever happened is evident" therefore "if it's not evident then it never happened").

Anyway that's how I reconcile ruben2020's answer ("from an inconstruable beginning") with Lanka's ("the cosmos is an imponderable"). I don't think it's meant as a cosmological or mathematical statement.

According to dependent origination I took birth (i.e I came into existence) due to ignorance.

Since when birth of ignorance has been happening?

Dependent origination is a cycle.

You cannot point out a starting point of a circle. There is always a point before a given point in a circle.

Likewise the main cause for the origination of ignorance has been mentioned in dependent origination.

Hence you cannot find the beginning of ignorance.

At Savatthi. There the Blessed One said: "From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on."

~ Assu Sutta, Anamatagga Samyutta, Samyutta Nikaya, Tipitaka

Very importand question! Why? Because it's required to see the meaninglessness of being a recidivist to actually change ones ways. Such a question and remembering the answer is not one to put aside but to repeat till it bring fruits, till one sees the recidivist clearly! (speech can cause so much... this or that and best.) The question and answer around it teaches not only the terrible size of suffering, but also gives when clear the right amount of effort to stop this foolish recidivist-hood.

Anamatagga-samyutta — The unimaginable beginnings of samsara

"From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries/the death of a father... the death of a brother... the death of a sister... the death of a son... the death of a daughter... loss with regard to relatives... loss with regard to wealth... loss with regard to disease/experienced stress, experienced pain, experienced loss, swelling the cemeteries/...blood you have shed when, being water buffaloes, you had your water buffalo-heads cut off... when, being rams, you had your ram-heads cut off... when, being goats, you had your goat-heads cut off... when, being deer, you had your deer-heads cut off... when, being chickens, you had your chicken-heads cut off... when, being pigs, you had your pig-heads cut off/...arrested as thieves plundering villages, you had your heads cut off... when, arrested as highway thieves, you had your heads cut off... when, arrested as adulterers, you had your heads cut off/... — enough to become disenchanted with all fabricated things, enough to become dispassionate, enough to be released."

[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]

The traditional answer is that you have been reborn countless times. Ordinarily, I do not offer my personal opinion on this website. But only because you asked, based upon my meditation on the evolution of consciousness, this traditional answer is accurate because a sentient being is born with the intelligence of a tiny biological organism. Only by being reborn into a symbiotic relationship with progressively more complex organisms does a being acquire human intelligence.

According to the Assu Sutta:

There the Blessed One said: "From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. What do you think, monks: Which is greater, the tears you have shed while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — or the water in the four great oceans?"

"As we understand the Dhamma taught to us by the Blessed One, this is the greater: the tears we have shed while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — not the water in the four great oceans."

"Excellent, monks. Excellent. It is excellent that you thus understand the Dhamma taught by me.

According to this article, the total amount of tears a person sheds on average in a lifetime is estimated to be 64 litres.

According to this USGS estimation and this article, there is an estimated 1.332 billion cubic kilometers of water in the Earth's oceans. That's 1.332 x 10²¹ litres.

If I divide that by 64, I get 2.08125 x 10¹⁹ lifetimes.

According to the short scale, that's 20 quintillion (20 x 10¹⁸).

Hence, according to the Buddha, every person has been reborn at least 20 quintillion times.

However, since the Buddha says that "a beginning point is not evident", I guess that it should be far greater than 20 quintillion times. Maybe a googleplex amount of times.