Disclaimer: When I say fastest growing set, I mean set with the fastest growing get-the-nth-member function. I don't know the technical term for this property and my math vocabulary is limited.
The Goldbach conjecture states that every even number can be expressed as the sum of two primes. Let's only concern ourselves with odd primes and even numbers >= 6. The growth rate of the nth prime number function ~ n*log(n). Anyway, I can't think of a set that grows more quickly than the odd prime numbers that still satisfies this property, or even another set that grows as quickly that isn't some trivial modification of the set of odd primes.