3, 31, 314159, and 31415926535897932384626433832795028841, are the first four terms in the "pi-primes" sequence: A005042. That 38-digit fourth term was attributed by Martin Gardner (in 1979) to Robert Baillie and Marvin Wunderlich. By 2000, a larger (fifth) term had yet to be found. That year, Clifford Pickover (under the guise of Dr. Googol) wrote in his "Wonders of Numbers" that there are likely infinitely many terms but "neither humans nor any lifeforms in the vast universe will ever know the next prime... It is simply too large for our computers to find." I wrote about Pickover's gross underestimation of computational progress previously and this serves as another example.
In 2001, Ed T. Prothro calculated that fifth term, composed of 16208 digits. In 2006, Eric Weisstein calculated the sixth and seventh terms, composed of 47577 and 78073 digits, respectively. In 2016, Adrian Bondrescu calculated the eighth term, composed of 613373 digits.
A fine point is that — as of this writing — the fifth to eighth terms are not proven primes, but probable only. That should not deter one from (pragmatically) thinking of them as primes.