There is no largest prime, and there is a really simple proof due to Euclid.
We will show that for any number n, there is a prime p, with p > n. Choose any number n. There are k primes less than or equal to n for some value k. Multiply them all together and then add 1 to get x = 2*3*5*...*pk + 1. x is not divisible by any of the primes, since it has a remainder of 1 when you divide x by any of them.
There are two possibilities. x may be a prime, in which case it must be greater than n, since it would be different from any of the primes less than or equal to n. Set p=x. On the other hand, x may not be a prime, in which case it can be expressed as the product of primes. All of these prime factors must be greater than n, since x is not divisible by any prime less than or equal to n. Set p equal to any of these prime factors.
We can choose n to be arbitrarily large, which means that the primes greater than n are arbitrarily large, meaning there is no largest prime.
As an example, find a prime > n=6. The primes <= 6 are 2, 3 and 5. x = 2*3*5+1 = 31. Since 31 is a prime, so we can set p = 31.