Basic hardcore skill set

Here's a list of 15 skills that I think are the most useful ones to perm in your first hardcore ascensions. Ultimately, this is my opinion, and you can probably see that I'm biased against combat buffs and towards passives. I tend to avoid the buff classes when I ascend. It's annoying to have to manage my mp out of combat, but if there's a really awesome combat buff out there let me know and I'll add it to the list.
Unfortunately there's no optimal ascension order for hardcore no-path, and I don't recommend playing 15 hardcore oxy ascensions - at least not before you've gotten these skills. I've starred the 7 that I find most useful, though.
Please let me know if there's any other skills that are considered part of the basic hardcore skillset these days, and I'll get the list updated.

+item/meat passives - if you don't need to equip a familiar such as a baby gravy fairy or leprechaun, you can use a +exp or hp/mp restoring one instead:
Expert Panhandling - Sauceror
Gnefarious Pickpocketing - Moxie sign zone
Nimble Fingers - Disco Bandit
Mad Looting Skillz - Disco Bandit
Powers of Observatiogn - Moxie sign zone

Crafting - for tasty stuff:
Advanced Cocktailcrafting - Disco Bandit
Pastamastery - Pastamancer

Familiar passives/buffs - passives are best, but buffs are useful for the NS quest:
Amphibian Sympathy - Turtle Tamer
Empathy of the Newt - Turtle Tamer
Leash of Linguini - Pastamancer

Random useful stuff - nifty things:
Ambidextrous Funkslinging - Disco Bandit
Overdeveloped Sense of Self Preservation - Disco Bandit
Torso Awaregness - Moxie sign zone

Healing/turn saving - in hardcore, it's nice to be able to heal or rest at the camp without worrying about how many turns you're wasting:
Disco Nap - Disco Bandit
Tongue of the Otter - Seal Clubber

Definitely solid choices for hardcore-permanent skills. I feel you missed some very important ones though.

Advanced Saucecrafting: Taken by itself, Saucecraftery allows you to create Sauceror potions, and almost every single potion a Sauceror can create is ridiculously powerful. Tomato juice, for example, is a +50% bonus to all stats. It's basically a free ride through a boss fight or 5 turns of owning a zone (10 if you're a Sauceror), and that's the easiest potion to make. Taken in conjunction with Pastamastery, it grants you the ability to make reagent dishes, which means using easy-to-find ingredients to make very, very effective food.

The Moxious Madrigal: Moxie is the saving grace for every class - the higher it is, the higher-level zones you can get to, which means more experience, which means a faster game that gets you through Hardcore more easily. +10 Moxie is amazing for Hardcore characters - it means adventuring in areas 10 levels higher than normal. And all for 2 MP per 5 turns, or 10 if you make yourself a Rock 'n' Roll Legend. Very powerful, very cheap buff.

Pulverize: Another multi-use skill. Smashing stuff into powder means elemental resistance, which means easier getting through the Daily Dungeon, completing the McLargeHuge quest, and beating Lord Spookyraven. Nuggets mean extra elemental damage when you need it. And while they won't come up until level 11 or so, using wads means more adventures, more stat gain, and a powerful buff for 10 turns. In addition, Myst classes can use the various ingredients to make a wide variety of chefstaves, and Musc classes get the Malus for wad access much earlier.

Also, it goes without saying that Pulverize is pretty much the most important skill to have if you're going for Oxy.

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In addition, there's a few more skills that aren't essential, but do make things much easier on a Hardcore runner.

Tao of the Terrapin: Take less damage. Period. Simple as that.

Wisdom of the Elder Tortoises: Like buffs? Cast more of them. Insanely useful for Myst classes.

Entangling Noodles: This one almost goes up in the top category, because stealing 1-3 free turns of beatdown on an enemy is amazingly useful. And it's cheap, too.