Most of your DNA in the body doesn't change except for B and T cells. These are the only types of cells that actively mutate their receptor genes via a protein called RAG1 and 2.

Their ability to randomly mutate and the body's ability to remove the ones that are bad are the basis for how our immune system can seemingly fight off most pathogenic diseases that infect the body.

Germ layer cells that produce sperm and egg produce cells that are genetically distinct from the hosts original (somatic) cells through recombination and cross over events during meiosis. I would consider this to be another genetically different cell type because the code has been shuffled (the linear sequence of DNA is no longer the same). Keep in mind that the large majority of the DNA is still identical to the host, it's just a few shuffled genes and alleles.

DNA repair over an organism's lifetime will result in mutations in the DNA of the damaged cell. cells that undergo mitosis frequently also mutate via mistakes made by DNA polymerase. I don't know the mutation statistic off hand but it happens. Most of the time these mutated cells will kill themselves via apoptosis or your immune system will kill the cells for you. The ones that get away with it are cancer cells that form tumors or stem cells gone rouge that form teratomas.

Virus and other genetic elements like transposons can also cause mutations when they jump around your genome.