Just one d6.
I don't see anywhere that it specifically defines "damage die" for a weapon, but the fact that you roll one extra die, then two and three, indicates to me that you should only roll one extra die (of same the number of sides as the damage die or dice of the weapon).
For a similar construction, see PHB 196 which describes rolling "the damage die or dice" of a weapon, spell, or harmful ability.
This is in contrast to 4e, which specifically says that the "W" of a weapon is equal to its total damage, so that "2W" would mean twice the number of dice.
5e is a game where the rules try to homogenize a lot of roles. Most classes either use strong armor or have easy access to non-armor AC of 13 + a modifier. Everyone has the same proficiency bonus at the same level. Et cetera. The designers have called the general concept here "bounded accuracy," but I think it applies to damage as well.
If you roll 2d6 each time, then the multiplicative effect means that by the end you'd be rolling an extra 6d6 in comparison to barbarians using most weapons, who would be rolling an extra 3dX. This doesn't seem consistent with the even capabilities shown elsewhere in the system.
From a realism standpoint, my interpretation of the rule would mean that while a Greatsword does a ton of damage all the time, you can't squeeze much more extra damage out of it. There's a limit to how much harm you can cause, even as a lucky and skilled barbarian.
Of course, this doesn't matter too much. It only happens on crit, and the expanded critical ranges come with Fighter, so it's only likely to be notably broken once you reach Fighter 3/Barb 9. 12th level is a good time to deal arbitrary amounts of damage.