0.625

This number is neither prime nor composite.

In "A Density Version of the Vinogradov Three Prime Theorem" Xuancheng Shao proves that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible. [Post]