Probability of finding a small number within a much larger one

[Hans Havermann]

Specifically, I've determined that 10000013 is the smallest length-8 
number not found embedded in the 369693100 digits of 9^(9^9). Could 
someone with a good grasp of probability make the case that the 
time-consuming search of length-7 numbers is not likely to yield a 
smaller result?

[The sequence (just) submitted is 2, 2, 1, 10, 108, 1065, 10699, 
1000005, 10000013, 2, ... Smallest positive-integer string not 
explicitly embedded (in a normal left-to-right fashion) in the full 
decimal expansion of n^(n^n).]