double
is a fixed-size type, typically 64 bits, with 53 bits of precision; so it can't accurately represent any integer with more than about 16 digits. The largest standard integer type typically has 64 bits, and can represent integers up to about 19 digits. 100! is much larger than that, so can't be represented accurately by any built-in type.
You'll need a large integer type, representing a number as an array of smaller numbers. There's no standard type; you could use a library like Boost.Multiprecision or GMP or, since this is a programming challenge, implement it yourself. To calculate factorials, you'll need to implement multiplication; the easiest way to do that is with the "long multiplication" algorithm you learnt in school.