I've always been curious what happens to artifacts that are discovered, but aren't of the teams interest. So let's say for example a site that has been occupied by Byzantines, Arabs and then Romans in Sicily finds multiple artifacts from those later eras but the team is only interested in Republican era settlements.
And a follow up question. Do teams ever did deeper than their projects aims. So let's say for example would a team that is interested in the Hellenistic era ever go down to the bronze age? Or are they only concerned with excavating down to a particular level?
If you do not get an answer here, I would suggest trying again at /r/Archaeology. This should be a somewhat tricky question to answer with AH's standards because there is not really a single standard, and it varies a lot based on what exactly you are talking about (for example, Roman pottery is basically all over the place and if you are digging an Iron Age site it will be treated differently than if you find Bronze Age pottery).