A factorion in base $N$ is a natural number equal to the sum of the factorials of its digits in base $N$. So, the decimal factorions are:
$1 = 1!$
$2 = 2!$
$145 = 1! + 4! + 5!$
$40585 = 4! + 0! + 5! + 8! + 5!$
It's easy to prove that, in any base $N$, there is no factorion $> d \centerdot (N-1)!$ where $d$ is the maximum number (of digits) such that $N^{d-1} \leq d \centerdot (N-1)!$ In other words, beyond $d$ digits, the number will grow faster than the sum of its digit factorials. Thus, any base can be exhaustively searched in a time proportional to this upper limit, at worst.
However, I'm trying to find an algorithm more efficient than iterating that many times. I feel like there must be a way to eliminate large branches of the search, perhaps by searching in non-numerical order, but I haven't found any yet.
Any reasonable pre-calculations are allowed, including a table of the factorials of the digits.