A bugbear's long-limbed trait says
When you make a melee attack on your turn, your reach for it is 5 feet greater than normal.
Suppose the bugbear has multiple attacks on its attack actions and takes them without intervening movement (or triggered reactions/bonus actions). Does the bugbear's reach reset to 5ft for an instant between attacks, or should it be regarded as 10ft for the entire sequence?
Why does it matter?
As an example, consider a bugbear with multiple attacks and the grappler feat, which gives it advantage against a grappled opponent.
Now, if a bugbear grapples at 10ft range, it's normally pointless: it can perform the grapple (which is a special attack), but once it's established the bugbear's reach reverts to 5ft, the enemy is no longer within reach, and the grapple ends immediately.
However, with multiple attacks and the grappler feat, it could potentially get advantage on attacks subsequent to the grapple, before the grapple is lost after the last attack. But this only works if the 10ft range is maintained during the sequence of attacks. If it reverts in between attacks, even for a moment, we're back to grapples ending the instant they're established.
Best Answer
RAW, the trait technically does not increase "your" reach
The trait states, emphasis mine :
Emphasis for the fact that what is increased is "your reach" only for this attack, and not your global reach. Meaning that if interpreted very close to the text itself, it is not your character's reach that is increased. Instead, the "reach" used for the attack, and only the attack, is increased.
Thus, any effect depending on your character's reach stat is not affected by this trait at any point, aside of any effect directly dependant on the attack that benefits from this range increase.
In that sense, even if the attacks of your sequence were to be perfectly following one another, your character's reach would technically be the same, meaning that the grapple would end directly after it started.
In terms of actually ruling this in a game, as a DM, I would definitely agree with Thomas Markov's answer and allow the grapple to hold for the duration of the attack sequence, as it seems a fairly original, if unoptimal, way to use attacks.