What genetic mutation would cause the most dramatic change in the genetic make-up of an individual?

The answer is insertion mutation.

Insertion mutation is a genetic mutation, which happens when one or more nucleotide base pairs are added into a DNA sequence. It is a type of mutation that would more or less cause the most dramatic change in the genetic makeup of an individual simply because by inserting an extra base pair you are disrupting the reading frame, thus affecting the primary structure of the polypeptide as it is being synthesized during translation.

The codons after the insert will all be different may or may not be coding for different amino acids.