
New record prime: 2^{74,207,281}1 with 22,338,618 digits by Cooper, Woltman, Kurowski, Blosser & GIMPS (7 Jan 2016). 

Question: Where can I find a complete list of the 106digit primes?
If you did not find that answer funny, then you definitely need to read the rest of this entry! How many 106digit number are there? Well, the smallest is 10^{105} and the largest is 10^{106}1, so there are 10^{106}10^{105} of them. There are nowhere this many atoms in the entire universe, so of course there is no such list of integers. What if we just want the primes? That does not help much. By the Prime Number Theorem the number of primes less than x is about x/log x where log x is the natural logarithm of x (roughly 2.3 times the number of digits in x). So the number of 106digit primes is about
Good luck fitting that list in this universe! So how big can x be and we still have a complete list? That depends on how much of the world's resources you want to dedicate to the list. If you want to stick with a single computer, then definitely less than 20 digits. You might, with proper compression techniques, create a list of all 16digit primes. 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 