The largest Fermat number with a known factor is $F_{3329780}$ with prime factor $193\times2^{3329782} + 1$
At googology are many examples of huge integers. In general, for any integer $a$, it's easy to list larger integers such as $a+1$, $a!$, or $a^a$. If some non-trivial operation or function is required, such as "requires a non-trivial prime factor", then perhaps it isn't so easy to grow them. I'll say $2 \times F_{3329780}$ doesn't count because we already know about those factors, therefore they are trivial.
"What is a good definition for a non-trivial operation or function?" might be a good part of this question. One example might be making a semiprime. The trivial method is to multiply two primes. Don Reble made a semiprime with an elliptic pseudo-curve and the Goldwasser-Kilian ECPP theorem. There is non-equivalency between Reble's method and $a \times b$.
Are there any known integers $G > F_{3329780}$ where results of non-trivial and non-equivalent operations on $G$ are known?
