Titanic prime is a prime number with 1000 or more digits. The name was conceived by Samuel Yates when he began to catalog the largest known primes in 1984. Only 110 titanic primes were known in 1984, the number being hundreds of thousands by now (Sep 26, 2001) and constantly increasing as computers keep crunching for larger and larger primes for cryptographic purposes. For example, the RSA cryptosystem keys longer that 8192 bits are products of two titanic primes.
Phil Carmody's "illegal" prime numbers are probably the most famous examples of titanic primes so far.
Thanks to Jongleur for the RSA info.