This is interesting. The largest prime number is: 2^74207281-1, which was announced just last month. It has 22 million digits in base 10.
The largest number which has been factored is:
2^383858863-1 which is divisible by 2385654186158054311
Someone is fascinated with the number 2^383838383-1, which has no factors below 2^83. This appears to be the highest probably prime number.
A lot of work is being done on primes in the range of 2^332,192,831 to 2^332,399,999. If one of these was proven prime, it would have 100 million digits in base 10.
The number 2^999999893-1 has been factored. It is divisible by 5999999359. That is the highest factored number listed in the Mersenne database, so it almost certainly is the biggest factored number.