If I use Metagross' "Meteor Mash" attack several turns in succession, does the damage keep piling up as 60, 120, 180, 240, etc.?
No, Meteor Mash tops out at 120. It only adds +60 to the base 60 damage, it does not accumulate on successive turns.
Source: Celestial Storm FAQ (2018-07-20), TPCi Rules Team (2018-08-02)