Fibonacci primes, primes of the form $2^n-k$ and beyond

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes $u_n$, or else there exists a constant $c_u>0$ (which we can give good approximations to) such that there are $\sim c_u \log N$ primes $u_n$ with $n\leq N$, as $N\to \infty$. We compare our conjecture to the limited amount of data that we can compile.