So I know that characters have a standard melee reach of 5 feet, which is for unarmed strikes and weapons like maces, shortswords, and mauls. I also know that weapons with the Reach property give an additional 5 foot reach, effectively increasing it to 10 foot reach for attacks with glaives, halberds, pikes, and whips. Now my question is would a Bugbear's Long Limbed feature still add another 5 foot reach, making it 15 feet? And on top of that, if you were to take the Martial Adept feat and select the Lunging Attack maneuver from the Battle Master subclass of Fighter, would you be able to achieve a 20 foot melee reach?

As far as I can tell, yes and yes.  Both increase the reach "for that attack" by 5 feet.

Bugbear doesn’t increase reach UNTIL you make an attack, nor does Lunging Strike. So no, the Bugbear racial doesn’t increase the zone of squares within their reach for Opportunity Attack purposes, though it DOES allow them to attack targets that are outside of their normal reach.

Both the Bugbear's ability and the Lunging Attack state they increase the reach by +5 ft during your attack.  They each definitely stack with a reach weapon and both appear to also stack with each other.  

I think a Bugbear Battle Master using Lunging Attack, with a Reach weapon does in fact have a 20 ft reach.

Right, so for one attack on your turn, you could have a 20 foot reach.

Yes, for opportunity attacks, the Bugbear has a 10 foot zone with a normal weapon 15 feet with a reach weapon. The thing to keep in mind is that if the target moves from 5 feet to 10 feet, the bugbear wouldn't get an opportunity attack like everybody else because the target didn't leave reach range (RAW).

Quote from Wysperra >>

Yes, for opportunity attacks, the Bugbear has a 10 foot zone with a normal weapon 15 feet with a reach weapon. The thing to keep in mind is that if the target moves from 5 feet to 10 feet, the bugbear wouldn't get an opportunity attack like everybody else because the target didn't leave reach range (RAW).

No, for opportunity attacks they only have a 5 foot reach with a normal weapon and 10 with a reach weapon, same as other races.  Long-Limbed says "when you make a melee attack on your turn, your reach for it is 5 feet greater than normal," however opportunity attacks are a reaction that trigger before you "make a melee attack."  Besides, opportunity attacks rarely come up on your turn.

Quote from Wysperra >>

Yes, for opportunity attacks, the Bugbear has a 10 foot zone with a normal weapon 15 feet with a reach weapon. The thing to keep in mind is that if the target moves from 5 feet to 10 feet, the bugbear wouldn't get an opportunity attack like everybody else because the target didn't leave reach range (RAW).

No, for Opportunity Attacks (whether on its own turn or another creature's turn) a Bugbear has the same 10 foot reach as any other character does with a reach weapon, or the same 5 foot reach as any other character does with a regular weapon. However, a Bugbear can make attacks at creatures that are 5 feet outside of its reach, by extending its reach for that attack. That doesn't help it make OAs or threaten further squares or expand its "within your reach" area for any other feature that might look for that reach prior to an attack being made. 

Opportunity Attacks

In a fight, everyone is constantly watching for a chance to strike an enemy who is fleeing or passing by. Such a strike is called an opportunity attack.

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

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I'm not seeing a 5 foot limit listed anywhere in there.

Quote from Wysperra >>

Opportunity Attacks

In a fight, everyone is constantly watching for a chance to strike an enemy who is fleeing or passing by. Such a strike is called an opportunity attack.

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

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I'm not seeing a 5 foot limit listed anywhere in there.

What I highlighted, "moves out of your reach," is the 5 foot limit.  Bugbears have a reach of 5 feet, same as all other playable races.  Only when they make an attack can they temporarily extend their reach, but opportunity attacks are triggered before you make an attack.

But Reach weapons add 5 feet to the opportunity attack range, thus with no other bonuses, anyone using a Reach weapon has 10 foot melee reach for all attacks, including opportunity attacks.

Bugbears will only get 15 when they make attacks on their turn. With Lunging Attack, they could potentially get 20 feet. Or with other options such as a Way of the Four Elements' Fangs of the Fire Snake, they could get and additional 10 feet, but only for unarmed strikes, making it a 20 foot reach for that attack.