"what are the largest values of n for which n! has fewer than 100 decimal digits

I can only thin of a hot and miss method to do this . That is using Stirlings approximation for n!-

ln n!  = approx   n ln n  - n

so if we try n = 60    this gives 4.2796 * 10^80 

so we could try higher values:-   

n = 65 gives an 89 digit number

n = 70 gives 98 digit number

n=71 gives me an memory error on my calculator!

Lets try an online calculator

Web2.0 calculator gives  a 101 digit number or n=71

70 is your answer