Again, there are many variations from state to state, so how Ohio works is likely quite different from here. Since the late 1970's, California has had a law which prohibits annual property tax bills from increasing by more than a fixed percentage (it followed some pretty steep increases during the 1970's which caused concern). But when a property changes hands, that triggers a reassessment of value, so if the previous owner had been there a long time, that can result in a substantial rise in assessed taxes. As you mention with certain improvements (an extension, adding a pool, whatever) that can also trigger a reassessment by the county, so understandably there are many properties around here which have had all sorts of additions carried out over the years without bulding permits. Unpermitted garage conversions to extra living space are very common, for example.
Almost lunch time here in the West. Happy Thanksgiving!