We have the partial factorization
$$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$
The first factor is $F_{24}$. It is composite, but no prime factor is known. A prime factor of the second factor must have the form $2^{25}\cdot 3k+1$. The second factor can be written as $F_{25}-F_{24}+1$
Is it known whether it is prime ?
If no, is a prime factor known ?