Bugbear - Surprise attack

If you hit a creature with an attack roll, the creature takes an extra 2d6 damage if it hasn’t taken a turn yet in the current combat.

I have had some people say that this happens every round, but I think it's only in the first round. What's correct? 

The answer is different depending on which version of the Bugbear species you're using.

For the version from Mordenkainen Presents: Monsters of the Multiverse, which is the one you quoted, it's usually only going to happen in the first round, because that's generally when you're going to be fighting creatures that haven't taken a turn yet. ("In this combat" means in this combat, not in this round.)

However, it is theoretically possible for it to happen again if, in a subsequent round, you attack someone who still hasn't taken a turn in this combat. That might happen, for instance, if your enemies summon reinforcements who don't arrive until the second round (or later). 

For the older version from Volo's Guide to Monsters, the feature is worded differently such that it explicitly only applies on your first turn and you can only use it once per combat.

Yeah, I did take it to mean that "any" enemy has not taken an action applies. Thanks

Quote from RighthAndyman618 >>

Yeah, I did take it to mean that "any" enemy has not taken an action applies. Thanks

I feel I should add that "not taken an action" is not the same thing as "not taken a turn". As it's written, you're not able to use this against someone who has taken a turn but wasn't able to do anything on that turn (due to being Paralyzed or whatever).

Quote from RighthAndyman618 >>

If you hit a creature with an attack roll, the creature takes an extra 2d6 damage if it hasn’t taken a turn yet in the current combat.

I have had some people say that this happens every round, but I think it's only in the first round. What's correct? 

 It probably could if you are dealing with multiple opponents, and some haven't taken (or been able to) a turn yet for any number of reasons.