Shortest strings not present in the human genome
The human genome contains less than 416 base pairs, so, even after including a factor of 2 for the two strands and another to allow for some genetic variation, there must certainly exist some 17-base string not found in either strand of the genome.

Now, 417 bits is two gigabytes, which fits easily within the memory of a typical desktop computer these days, so a simple and efficient algorithm for finding such a string would be to set up an array of 417 bits, initialize them all to zero, and then simply iterate over the entire genome (including all known variations), taking each 17-base subsequence (and its complement), mapping them to base-4 integers and setting the bits in the array at the positions given by those integers to one.

Once you're done, any remaining zero bits will correspond to strings not found in the genome.  More to the point, the longest consecutive block of zero bits will give you the shortest base string not present in the genome.