There is a joke that when one counts something in linear algebra, the only possible results are 0, 1, and infinity.  What about in number theory?  Well, if we count the primes, we get infinity.  To get a finite number, we could try to count primes less than some fixed N.  It was proven in 1955 that something interesting happens sometime before N = 10^(10^(10^1000)).  Huh?!  What if we count gaps between primes?  Just in 2013, it was proven that something interesting happens for gaps less than 70,000,000.  What if we try to write down a really big prime number?  Right now, nobody can write down a prime bigger than 10^(10^8).  In this talk, we'll introduce some of these big numbers in number theory, we'll carefully say what they mean, and we'll describe progress on making the first two big numbers smaller and on making the third big number bigger.