[RPG] Extra Attack + Horde Breaker = 3 attacks, total

Extra Attack only modifies a regular Attack action

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

I think the fact that "Attack" is capitalized seems to implicate the Attack action under actions in combat vs. any attack.

Horde Breaker could be used in conjunction with Whirlwind Attack

Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.

Since Horde Breaker triggers off a "weapon attack" not an "Attack action" you should be able to use Horde Breaker to target one of the enemies next to you. Its simply triggering off one of the attacks made as part of the whirlwind.

Optimization breakdown

Since all Hunters get either Volley or Whirlwind, the Extra Attack class feature is there as a stopgap in the meantime and also a fallback for when there is only one enemy in range to be targeted. Horde Breaker should still be viable even though its utility will diminish in comparison upon reaching level 11 as a ranger, but even then its still one more attack per turn.

A dual wielding melee ranger may take up to 4 attacks, a ranged ranger or a ranger who is single wielding may take up to 3 attacks.

Horde breaker allows you to make an extra free attack when there is an enemy within 5' of an enemy you attack weather you are making a melee or ranged attack. Extra attack allows you to take an extra attack as part of your attack action. Dual wielding allows you to make an attack as a bonus action when you take the attack action. So there are two distinct situations:

Dual Wielder:

• Action: 2 attacks
• Free: Horde Breaker attack
• Bonus Action: 1 attack

Single Wielder/Ranged

• Action: 2 attacks
• Free: Horde Breaker attack

So the correct answer to your question is 4 attacks.