A number of the form $b^{n}\pm 1$ is a Cunningham number, with non-square integer $b>1$ and integer $n>2$. The Fermat and Mersenne numbers are Cunningham numbers. The Cunningham Project records various results.
For $n=2$, the largest twin primes likely provides the answer, $(3756801695685 \times 2^{666669})^2 - 1$ in this case.
Is $F_{11} = 2^{2048} +1$ (617 digits) the largest completely factored Cunningham number? Or are there some special numbers of this type that are completely factored?
EDIT: I'm going to also rule out Wagstaff primes. $2^{13372531} +1$ is (probably) completely factored.