The spell Melf's Acid Arrow does 4d4 on hit and 2d4 next turn (I'm ignoring what happens for a miss). If you roll a 20 for your spell attack roll, you double the 4d4 to 8d4.
What about the 2d4 next turn? Does that get doubled too or does that just stay at 2d4 because it occurs on the next turn?
I know that How does extra damage work for critical hits? covers additional kinds of damage during a critical hit, but this is the same substance that did critical damage, continuing its damage in a later round.