My question is, with each new starter I do, is that considered a new generation of yeast? If so, about how many generations (or cycles?) can I do this, considering my sanitation is good?
Several generations of yeast cells are created every time a culture is propagated. However, we can conveniently think of each subculturing event as a generation.
There are no hard and fast rules with respect to how many generations a yeast culture can be subcultured. A yeast culture may remain stable for as little a couple of generations or as long as thousands of generations. I often suggest that people who want to maintain a liquid culture do so until it starts to exhibit a decline in performance. Yeast culture performance decline usually occurs over more than one generation.
If a brewer repitches a culture long enough, it will adjust to his/her brew house. Many house strains started out as a common strain and became house strains through mutation. The house culture that Ipswich was using back in the nineties started out as Wyeast 1028. I do not know if they are still using the same house culture.