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A Gigaprime is a prime number whose decimal representation has 1,000,000,000 or more digits. There are infinitely many gigaprimes because there are infinitely primes. In fact, since there are only finitely many natural numbers with less than 1,000,000,000 digits, "nearly all" primes are gigaprimes.
As of this writing, no gigaprimes are known explicitly (nor has one of even 100,000,000 digits). The EFF is offering a 250,000 USD reward for the discovery of the first gigaprime.
Chris Caldwell has proposed the alternate name bevaprime.