[math-fun] Prime question

Let f(K) = 2K+1. Set f_1(K) := f(K), f_(n+1)(K) := f(f_n(K)). ---------------------------------------------------- QUESTION: What is the sup of all n such that there exists a prime number p for which f_r(p) is prime for all r in the range 1 <= r <= n ? In particular, could this sup be infinite? ----------------------------------------------------- (It's easy to see that f_r(K) == (2^r)(K+1)-1.) Of course the same question could well be asked of any other function taking positive odds into themselves, e.g., linear functions of form K |-> A*K+B such that A+B is odd, as well as tons of polynomials. Is much known about these? --Dan