Need a hyper-defensive opening for black.

Title. I need a hyper-defensive opening for black, preferably after 1. e4 and working for higher ELOs. I'm leaning towards the Hippo defense right now. Halp?

Karo-cann is a good defense

and maybe Sicilian defense 

The Hippo, the Pirc, the Hedgehog, these are all very fine. The Modern with 1...g6 is great too. You can aim to put the knights on e7 and d7, Hikaru does this a lot.
Watch some of Karpov's games, or Petrosian's, they have amazing defensive setups.

I wouldn't recommend the Sicilian as things get tense soon - little time for maneuvering before attacks and sacrifices start opening the lines. But if you do I advise the Scheveningen for its defensive merits. Castle early and maneuver the knights to the queenside a la Kasparov. It can also transpose from the Najdorf and you don't have to castle there.

Defenses like the Hippo, Pirc, Modern, etc, do not work at higher ELOs.

There are basically 4 options, all of which have sub-options within them:

1...e5

French

Sicilian

Caro-Kann

Which is best among those 4 is strictly a matter of taste for each individual person.  It could be 1...e5 for one person, the Sicilian for another, and the French for another, and none of the three would be wrong!

You want to work for higher Elo?
That means, you need to learn more.
Then you want practice in tactics.
That means: rather than playing "hyper-defensive" chess, you should seek out tactical openings.
When your opponent plays 1.e4, respond with e5.
The Open Games that ensue will have plenty of tactics. You'll learn the trix of the trade.
And keep hitting up the Puzzles, to hone your tactical skills.
That's how your rating will improve; not by choosing the ideal opening.

How do you define hyper-defensive ? If you mean hard to beat, ie larger % of draws compared to wins/losses, pick one with high draw % the from the 4 options ThrillerFan listed. That's probably the worst approach to take though. Better to try them out and intuitively your heart will know where it belongs, for important choices, the letter N from personality types is probably the way to go, aka trust your gut.

Worth noting that Petrosian, whom another user mentioned, was notoriously hard to beat and at the same time with the Black pieces he played a lot of King's Indians and Najdorfs ( as well as French & Caro ofc ), which is not exactly what comes to mind right after the word "hyper-defensive", but there you go.

Also worth noting that both Karpov and Petrosian were extremely good at calculation, which is why their defences were not breached.

1... g6 against anything except 1. b3/b4. The Modern often transposes to other Modern Defenses or other openings entirely (such as the Hyperaccelerated Dragon Sicilian, Gurgenidze Caro-Kann, Pirc, KID et cetera) It's definitely hyper-defensive, but some of the transpositions are anything but. I think it's one of the best openings due to the sheer flexibility you get, and it's essentially as viable against 1. e4 as the Caro-Kann or the French (the Sicilian and 1... e5 being slightly better than anything else in that regard).