Right now, the prime:
835738017*2^1290000 - 1 (2012 July 14)
is ranked 5350 on Caldwell's list, and the prime:
2400254545845*2^1290000 - 1 (2015 December 8)
is ranked 3349 there.
Of course almost all the primes in the list in between these two are also from this project. (One user, p199, seems to start from two primes found by this project, say k*2^1290000-1 and j*2^1290000-1 (with k<j), and then see if i*2^1290000-1 is also prime, where i = j+(j-k) = 2j-k is the next member of the arithmetic progression k,j,...)
If we are lucky that one of our finds will be the lower member of a twin, or a safe prime, or a Sophie Germain prime, our exponent 1290000 is still large enough to ensure that both that prime and the associated prime are in Top 5000 in their own rights.