For more on this, see KConrad's comment https://matheducators.stackexchange.com/questions/16636/is-t... and kcrisman's comment https://matheducators.stackexchange.com/questions/16636/is-t... on the post linked by your parent.
Also, the series 2^(2^n)+1 grows extremely quickly; the next number is 18446744073709551617 = 274177 * 6728042131072, and after that 340282366920938463463374607431768211457 = 59649589127497217 * 5704689200685129054721.
For Fermat numbers 2^(2^n) + 1, it is well known that every prime factor should have the form k 2^(n+2) + 1. The sequence (A007117) thus goes like: k = 5 for n = 5 (Euler in 1732, very managable), k = 1071 for n = 6 (Thomas Clausen in 1854) and k = 116503103764643 for n = 7 (only known in 1970 after the modern computer).
π(2^16)=6542: if you told me in elementary school that I might disproving something famous I can see myself working on that for a couple of hours daily for a week or so.
This succeeds for a little while, since (the product of no primes) + 1 is 2, 2 + 1 is 3, 2·3 + 1 is 7, and 2·3·7 + 1 is 43, and most math geeks will recognise these as primes. However, it fails immediately after: 2·3·7·43 + 1 is 13·139. (If you take "all the primes so far" to mean not just "all the primes generated this way" but "all primes less than or equal to those generated this way", then we get a slightly different trajectory: after 7 comes 2·3·5·7 + 1 = 211, which is prime; and then the so-called primorial (https://en.wikipedia.org/wiki/Primorial) 211# has a successor that is huge, way bigger than 2297, but is divisible by 2297.)
It is possible to modify the sequence so that it proveably generates all primes; see, for example, Booker - A variant of the Euclid–Mullin sequence containing every prime (https://arxiv.org/abs/1605.08929). According to Wikipedia, the notion was introduced in Mullin - Recursive function theory (A modern look at a Euclidean idea) (https://doi.org/10.1090%2FS0002-9904-1963-11017-4).
EDIT: Replying to child (https://news.ycombinator.com/item?id=20017324 )—sorry for doing it here; I'm "posting too fast", but don't want to ignore it:
> I'm genuinely confused by this. I don't see how 2·3·7·43 is "the product of all the primes so far", since 11, 13, 17, 19, 23, 29, 31, 37, and 41 are missing.
> Does the author mean "all the primes generated by the sequence", or "all the primes up to a certain number"? I always thought the Euclid method involved the latter.
Just to be clear, I'm not the author of the linked post. I did write "all the primes so far" to mean "all the primes generated in this way", which is the sense in which Mullin used it in the article linked above. Both approaches work to generate 'new' primes, in the sense that they're not among the primes multiplied together; but you must use the primorial if you want to generate new, bigger primes. I edited to take into account the more natural interpretation (see the parenthetical above), but probably it was while you were posting.