I am working with recursive prime sequences of the form

Prime[Prime[Prime[Prime[n]]]]

Unfortunately, the values get large fast, and the Prime[n] function tops out at around Prime[8trillion].

Is there a way to get more primes, such as a Mathematica package or? I haven't been able to find any complete lists of primes larger than this. Ideally, I need a complete list of primes up to Prime[10^100] or so.

I suppose I could let a Sieve of Eratosthenes run, and dump primes to a text file, but I'm guessing that would take... months or years to get to Prime[10^100]? Any advice is appreciated.