This would continue, and the enzymes becoming more accurate (other factors other than enzymes can be considered, e.g. the solution in the nuclear envelope, but I'll just stick to enzymes) until the probability of mutation is incredibly small.
This also offers a small explaination to the dinosaurs, because if they had less DNA mutations then they would be less likely to survive a major climate change.
Then it sort of reaches an equilibrium, then, doesn't it? Where members of a species become so homologous and clonal that a single disease or something wipes them all out - except for the mutants in the population who didn't inherit the "perfect" polymerase sequences, etc.
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