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If I'm a Battlerager barbarian (from Sword Coast Adventurer's Guide, p. 121) and make 3 attacks on my turn while using Reckless Attack and raging, do I get 5 temporary HP total, or 5 temp HP per attack (15 HP total)?

The latter seems a bit high to be able to get per round combined with resistances.

The Battlerager's Reckless Abandon feature says:

Beginning at 6th level, when you use Reckless Attack while raging, you also gain temporary hit points

The Reckless Attack feature says:

Starting at 2nd level, you can throw aside all concern
for defense to attack with fierce desperation. When you make your
first attack on your turn, you can decide to attack recklessly. Doing
so gives you advantage on melee weapon attack rolls using Strength
during this turn, but attack rolls against you have advantage until
your next turn.

Since Reckless Attack only triggers when you make your first attack on your turn, it can only be used once per turn (and only on your turn, unlike Sneak Attack). Therefore, you can only gain the benefit of Reckless Abandon once per turn.

Also, as Purple Monkey points out, temporary HP from multiple sources does not stack, so even if you could use Reckless Abandon multiple times on your turn, you'd still only have 5 Temp HP: How does Armor of Agathys interact with getting temporary hit points?

It doesn't appear to be, since the ruling is that you can only use Reckless Attack once per turn, and you can only benefit once from your Reckless Abandon. Overall, Divibisan's answer is right in this situation.

However, I may have to ask about what you're trying to go for here. Perhaps you're trying to make something tanky, because the Battlerager isn't exactly the best subclass for the Barbarian when it comes to tank damage, or if you're trying to go for damage, because again, Battlerager isn't good for that either. Now, if you want an option to be tanky, go for the Ancestral Guardian Barbarian, as this ability really comes into play for that.

SPIRIT SHIELD
Beginning at 6th level, the guardian spirits that aid you can provide supernatural protection to those you defend. If you are raging and another creature you can see within 30 feet of you takes damage, you can use your reaction to reduce that damage by 2d6. When you reach certain levels in this class. you can reduce the damage by more: by 3d6 at 10th level and by 4d6 at 14th.

This would mean that you can use this every reaction (10 reactions) thus preventing up to 240 points of damage throughout your rage, up to Level 15 where you can rage for as long as you want, and I quote.

PERSISTENT RAGE Beginning at 15th level, your rage is so fierce that
it ends early only if you fall unconscious or if you choose to end it.

This would mean that the time limit no longer exists, so this would really be a big boost towards being tanky.

Or, if you're looking for damage, go for the Berserker subclass instead, and I quote:

FRENZY Starting w hen you choose this path at 3rd level, you can go into a frenzy when you rage. If you do so, for the duration of your rage you can make a single melee weapon attack as a bonus action on each of your turns after this one. When your rage ends, you suffer one level of exhaustion (as described in appendix A).

And this ability certainly helps with the damage part.

Retaliation Starting at 14th level, when you take damage from a creature that is within 5 feet of you. you can use your reaction to make a melee weapon attack against that creature.

So kindly explain to me why you'd want to take up something that could be improved instead of something that really does the job well.

And I apologize for ranting.