I'm a little uncertain on what you're asking. Are you asking about a hypothetical family that stays in one location without that location coming under non-Roman rule, or could they move to stay under Roman rule?

The Roman Republic goes back to 509 BC (the Roman Kingdom allegedly goes back to 753 BC), and they ruled Rome until 476 AD, for 985 (or 1229, if including kingdom) years.

The Romans ruled Corinth from 146 BC until the Franks under the auspices of the Principality of Achaea conquered it in 1210, or 1356 years. (However, there were some periods of Corinth's Byzantine history that are pretty obscure, so it's hard to definitely say Roman rule was continuous.)

If you allow moving to stay under Roman rule, then you could go all the way from the Roman Republic (509 BC) or Roman Kingdom (753) to the fall of Constantinople (1453) (1962 or 2206 years).

(It is possible that there was some area under continuous Roman rule for longer than Rome or Corinth, but I don't know of which. I almost included Trebizond as being under continuous Roman rule from 65 AD to 1461, 1396 years, but I decided that there were too many issues with their brief post-Manzikert occupation by the Seljuks, Theodore Gabras's treatment of it as a personal possession rather than a part of the Roman empire, and the issues of the legitimacy of post-Fourth Crusade Byzantine successor states in general to call it continuous Roman rule.)

A "short" generation would be about 13-14 years. Since you're asking about "at maximum," the maximum number of generations would be with the shortest generations. (Menarche over much of this time period would likely have been later than the 12-13 that 13-14 year generations imply for most girls, but there were exceptions, and you did ask for maximum.)

Without moving -- staying in Corinth, say -- that means a maximum of about 100 generations. If you allow moving, you could have up to 150-170 generations.