http://en.wikipedia.org/wiki/Budding
Even with normal binary fission, there is never an equal distribution of products between cells. Metabolite concentration, organelle distribution, etc. will vary. (If you think about it, it's kind of an interesting period of rough perturbation made to an extremely high-order dynamical system.)
At a kind of base level, though, the genome itself regularly experiences abnormal partitioning that leads to certain interesting states.
It's not strictly binary...
But back to the previous topic. Epithelia and endothelia are regularly dividing cell types. (It's interesting that most cancers arise from these lineages, right?) Blood cells are another lineage that undergo pretty frequent proliferation.
(On an entirely unrelated note, I can't wait for an HN discussion pertaining to quantum effects on biological systems--it's a topic I really want to learn more about. The absorption spectra discussions in the retinal thread has awoken my curiosity. For instance: though it's kind of a controversial, maybe even a pseudo-scientific topic, the quantum effects on microtubules are fun to think about and are still actively researched.)