Consider the following sequence:
$$ T_{1} = a,\: T_{i+1} = F_{T_{i}} $$
where $ a \in \mathbb{P} $ and $F_i$ is the $i$-th Fibonacci number.
Is there a value of $a \neq 5$ such that this sequence generates only primes?
I certainly don't expect an answer in the positive—as it would prove that there are infinitely many Fibonacci primes, which is an open problem—but I would like to know if there are any straightforward/obvious reasons for which such an $a$ wouldn't exist.