I seem to remember reading that embryonic cells will frequently replicate the section of their genome containing rDNA by splicing in duplicate genes. The cells use this to produce ribosomes at a rate which would be impossible with the DNA present in cells from full-grown organisms. Is this real? Are there other examples of cells altering their their own nucleotide sequence?
One case where human cells alter their DNA occurs in the immune system. In the early stages of the continuing production of B-cells and T-cells, the developing cell recombines (shuffles) genes in a particular DNA region which codes for what are called Variable, Diverse, and Joining gene segments. This shuffling is called somatic recombination or V(D)J recombination. This rearrangement gives the cell a genetic template for a random instance out of a enormous set of possible antigen-recognition proteins used by the immune system to recognize foreign molecules (in order to fight infection).
I'm assuming you mean, physically changing the DNA polymer. The answer is yes. And how they do this depends upon which cells they are and what they are supposed to do. A partial list:
In multicellular animals, cells DNA 'ages' where the telomeres, sequences at the ends of the chromosomes, will be degrade and shorten. This relates to how many times the cells have divided and acts as a sort of clock for development and probably also ageing. Telomere degradation does not happen in all cells - the germline cells are not modified of course. This is one of the probable reasons that cloning animals sometimes doesn't make perfect copies.
Cells will also add Methyl groups to the DNA nucleotides. Methylation is triggered when environmental or developmental conditions. The textbook example for methylation is the markers on the DNA of children whose parents have experienced starvation conditions. Methylation can disappear in subsequent generations. It also happens to cells in the brain which mark different stages in their brain development.
Meiosis, When sexual mating happens, the cells that make the sperm and eggs shuffle the pairs of chromosomes, editing so that about half of the resulting single set of chromosomes contain half of each parent's chromosomes.
Then, there is DNA repair. When radiation or chemicals breaks the chromosome, there are enzymes that repair them. Usually they read the opposite strand of the DNA to see how the repair happens. Sometimes the DNA is not reparable from the strand, and the result is not the same as the DNA the cell started with.
Short repeat DNA seem to show up in chromosomes - segments of short repeats can change length (AGGAGGAGGAGGAGT -> AGGAGGAGGAGGAGGAGGAGTAGGAGG etc). I'm not sure how this happens, but its different between identical twins. It also probably happens to cells in different parts of your body.
What else have I forgotten everyone?
There are also enzymes that coil the DNA up onto nucleosomes, but don't really make any changes, but we won't count that.