...Eek. That is a large number.

You could try "guessing" at values for the answer, then using that guess to put a lower bound and upper bound. For example, if \(\displaystyle S(S(k)) = 2\), then \(\displaystyle 10^9 \le k \le 10^{99} - 1\).

Don't know if that method'll work but it's the only feasible method I see...problem is you'll have to determine between which two bounds \(\displaystyle 2012!^{2012!}\) lies between. Have fun with that.