Q: What is known largest prime number with known index?

Once such an (index, prime) pair is known, it is easy to surpass it simply by iterating 'next prime' and adding the number of iterations to the index. So, perhaps a better question is for what power of two (or ten, or whatever) do we know the exact number of smaller primes. For two, as of 18 Dec 2014, it is 86. For ten, as of 2 Dec 2014, it is 26: http://oeis.org/A007053 http://oeis.org/A006880