Let $F_n$ denote the $n$ th Fibonacci number and define $f(n):=F_{F_n}$
$f(n)$ is prime for $n=4,5,7$
If we have $n>4$ and $F_n$ is composite , then we only have to know a prime factor of $F_n$ , say $q$. Then we have $F_q\mid f(n)$
The next few primes $n$ (after $n=7$) for which $F_n$ is prime as well (so $f(n)$ has a chance to be prime at all) are $$11,13,17,23,29,43,47,83,131,137$$
The small prime factors are ("?" means that no prime factor is known) :
11 1069
13 139801
17 11144839714262653068143078433637
23 ?
29 ?
43 ?
47 ?
83 ?
131 ?
137 ?
$f(43)$ is already huge (more than $90$ million digits) , so it is almost surely composite. Who knows prime factors of the seven unknown cases ? Is the prime factor of $f(17)$ the smallest ?
$f(23)$ and $f(29)$ are known to be composite. $f(23)$ has probably no prime factor below $10^{20}$ and for the other cases , there is no prime factor below $2\cdot 10^8$.
Who knows prime factors for the open cases ?