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I am theorycrafting a level 10 character. (If my math here checks out, there is actually a better build I could do for it at higher levels.) In this build we'll be taking 5 levels in Ranger and selecting the Hunter subclass, with the Horde Breaker feature chosen at level 3. This feature reads as follows:

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.

Now from what I've checked of the fighter, the subclass you pick for it doesn't matter all that much, and since none of them seem to really effect the maths I'm working with, we'll just say it's a Battle Master, since that's likely what I'd pick myself in the long run.

At level 5, rangers get their Extra Attack feature. By level 5, fighters have both their Action Surge (level 2) and first Extra Attack (level 5) features. Action Surge allows you to take another action once during your turn, regaining the ability to do so after a short or long rest.

Now if I am correct, Action Surge also allows for your extra attacks to be used if you take the Attack action with your extra action. (It may well also make it possible to retake a bonus action activated by any feats based on Attack actions, but I don't believe this to be the case.)

The Crossbow Expert feat (PHB, p. 165) has 3 benefits, two of which are going to be a part of this calculation. The first is that you ignore the "loading" property of any crossbow you are proficient in (the Light Crossbow is a simple weapon, while the Hand Crossbow and Heavy Crossbow are martial weapons; bear this in mind when picking classes/backgrounds). The second relevant feature is written as follows:

When you use the Attack action and attack with a one handed weapon, you can use a bonus action to attack with a hand crossbow you are holding.

Given all the previous information, I believe the following turn I am about to post is possible, provided you target an enemy that has another enemy within 5 feet of it. It is also important to mention that due to the rules behind ammunition, you will require both enough ammunition available for this and one hand free to reload your weapon:

• 
  Attack, Extra Attack, Extra Attack, Horde Breaker attack (I believe this can be taken at any point during your attacks, though I may be incorrect), bonus-action attack (via Crossbow Expert), attack (via Action Surge), Extra Attack, Extra Attack
  

This by my count is 8 attacks in a single turn, accomplishable once between every short or long rest. Assuming the use of the hand crossbow (as I am not sure how the light and heavy crossbows would work in regards to having one hand free to reload, although I imagine that depending on DM ruling it would be possible with them) this results in 8d6 piercing damage in a single turn at level 10 as a martial with only the Crossbow Expert feat.

This is likely not a majorly impressive feat and I'm sure it's possible to get more damage out with far less "faffing around" as it were, and I believe you can get much more out of it via the Sharpshooter Feat, but I am interested to see if my math checks out here.

I'm also interested to know whether you'd get a second bonus attack via Crossbow Expert as part of your attack action via Action Surge, as the feat itself does say, "When you use the Attack action and attack with a one handed weapon, you can use a bonus action to attack with a hand crossbow you are holding."

What is the maximum number of attacks I can make with this character in one turn, using Action Surge?

You'd have a maximum of 6 attacks

The Extra Attack feature of various classes do not stack, per the multiclassing rules, so your turn would look like this:

Attack action: 2 attacks

Action Surge (another Attack action): 2 attacks

Horde Breaker: 1 attack (if a valid target exists)

Bonus action (Crossbow Expert): 1 attack

Most of the time, though, you're going to want to cast and move Hunter's Mark to add an additional 1d6 per shot. I further recommend taking the sharpshooter feat so that you'd deal 2d6+10 damage (in addition to your Dex modifier) per shot.

And wear heavy armor. The DM is going to come after you for your trouble.

I have been informed that due to the multiclassing rules on class features preventing you from taking extra attack from multiple sources, at level 10 with 5 Ranger and 5 Fighter you would be able to perform only one Extra Attack, plus an extra extra attack following an attack action from your action surge. This changes the turn to the following:

Attack, Extra Attack, Hordebreaker, Bonus Attack (Crossbow Expert), Attack (Action Surge) Extra Attack

Which with the Hand Crossbow (Allowing 1 hand free for reloading, may be possible with the 2-handed crossbows, unsure of this) means that once per short rest you can hit 6d6 (plus any other bonus damage via feats and such), while being able to perform 4d6 every turn provided two enemies within range of your crossbow are within 5ft of eachother, and only 3d6 beyond that.

This does open up the class somewhat as you can take up to two levels away from Ranger to put elsewhere. Thank you for the information everyone, you've truly been a great help.

Firstly, there are several class feats that do not "stack", when multiclassing. The one that applies to you, is Extra Attack.

If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does).

So at level 5, Both Ranger and Fighter get an Extra Attack, but only one of these applies.

So, at this stage, you have access to an Attack, and an Extra Attack.

Secondly, technically it acts as a separate "attack", Horde Breaker works off using an attack action on your turn, but only once per turn (emphasis mine):

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.

Note however, that it does not specify that it needs to be a successful weapon attack. So, now, you have an attack, an Extra Attack, and Horde Breaker.

Crossbow Expert allows you to take an attack in your bonus action, provided you have a loaded hand crossbow (PHB p. 165, emphasis mine):

When you use the Attack action and attack with a one-handed weapon, you can use a bonus action to attack with a loaded hand crossbow you are holding.

In this set up, using this feat to achieve the additional attack on your bonus action will only work with one one-handed crossbow, as you need a free hand to load it for each attack. You can load it as part of the attack action, but you require a free hand in order to do so.

So, provided you are using a one-handed crossbow, you have an Attack, and Extra Attack, Horde Breaker, and use your bonus action to make another Attack, using Crossbow Expert.

And finally, Fighters at level 5 have access to one use of Action Surge per short or long rest.

On [y]our [t]urn, you can take one additional action on top of your regular action and a possible [b]onus [a]ction.

Once you use this feature, you must finish a short or [l]ong [r]est before you can use it again.

So, if you take an Attack action when you use your Action Surge, this opens you up to another Extra Attack (this does not allow for a second bonus action - just to clarify).

So, the final count is, that when you use all your feats in one turn, you have a total of 6 attacks on your turn. - Attack, Extra Attack, Horde Breaker, Bonus Action, Action Surge (Attack, Extra Attack).