Bitefinder

Some classes (and dedications of) grant unarmed attacks:

• Barbarians can gain 1d10 attacks through Animal Instinct while raging. These increase to 1d12 at level 7.
  However! If you can get a 1d8 unarmed attack some other way, you might be better off taking an Instinct with a higher flat damage increase instead.
• Magus has Arcane Fists¹ for 1d6 lethal fist attacks.
• Witch has Witch Armaments¹ for a few unarmed attacks, notable options being Living Hair (1d4 Bludgeoning with Disarm, Trip, Agile, Finesse) and Iron Teeth (d8 Piercing).
  If weird AP feats are allowed, hair can be later upgraded via Syu Tak-Nwa's Deadly Hair⁶ to 1d6 with added Grapple, thus replacing most uses of a free hand as far as athletic manuevers are concerned.
• Monks get some stances that do 1d8 damage without requiring you to be unarmored. Crushing Grab² is also cool if you're doing athletics.

• Clawdancer² grants a stance for 1d6 Slashing / 1d8 Slashing unarmed attacks with feats for added options.
• Rivethun Invoker² gets a Claw⁴ feat with 1d8 Spirit damage with Agile, Finesse, and Magical traits.
  However, this attack doesn't inherit your weapon proficiency runes, so it scales pretty poorly.
• Stone Brawler² grants a stance for 1d8 Bludgeoning unarmed attacks with Forceful and Magical traits.
  However, this dedication is lock-in and only has a single feat to choose from at levels 4 and 6.
• Dragon Disciple² (kobolds only) has a Claw⁴ feat for a 1d6 Slashing claws attack with Agile and Finesse.
• Sterling Dynamo² (uncommon) gets a few options (1d6 Bludgeoning, 1d4 Bludgeoning) with Grapple/Trip and an option for dice increment at cost of spending 1 action to wind up the dynamo.
• Wrestler² doesn't give you any new attacks, but has a bunch of Athletics-related fun things to do
• Spirit Warrior² increases your fist die size to d6 and gives it a Parry trait, with a later option to have +2 AC from parrying with your fist.
• Mind Smith² allows you to create a variety of weapons (notably one-handed 1d8 and two-handed 1d10 with Reach) and then add Grapple / Trip to it.
• Pactbound² (rare) offers an option to get 1d8 Bludgeoning horns⁶ with Shove and Trip traits. And on level 16 you can become Large and get Reach...
• Living Vessel (rare) offers a 1d6 Agile Finesse attack with damage type of your choice, and later offers constriction
• Ursine Avenger offers 1d8 Piercing and 1d6 Slashing (Agile) attacks while in its form.
• Thlipit Contestant gives you a 1d4 Bludgeoning (Grapple, Reach) tongue/tail attack in Flail group.

And here are some dedications that are probably not worth taking just for an unarmed attack:

• Vampire² (rare) gets Fangs (1d6 Piercing with Grapple) and has a Claw⁶ feat for 1d4 Slashing with a free grapple if you hit the same enemy twice in one round.
• Zombie² (rare) gets a jaws with 1d8 Piercing, but you can only bite a creature that you grabbed.
• Ghoul² (rare) gets Claws (1d4 Slashing with Agile, Finesse) and Jaws (1d6 Piercing with Finesse)
• Werecreatures² (rare) offer a variety of unarmed attacks with some drawbacks

Permanent:

• Grafts!
  Kind of expensive; might not be your vibe
• Tooth And Claw Tatoo (invested, level 6, 250◉)
  Essentially gives you a 1-minute Versatile effect for your best unarmed attack for 1 action, so if you regularly find yourself thinking "man, I wish my horns dealt slashing damage to these zombies", that's an item for you.
• Flesh Gem (invested, level 5, 160◉)
  Gives you a 1d6 Piercing attack. Not the best unless you need Piercing damage badly.
• Clawed Bracers (invested, level 7, 325◉)
  Once a hour gain a claw attack with 1d6 Slashing (Agile, Finesse) damage, and a climb speed!

Consumables:

• Mana-Rattler Liniment (consumable, level 9, 110◉)
  Gives a fangs attack with 1d6 Piercing + 1 poison damage for an hour.
• Fury Cocktail (consumable, level 4+)
  Gives you either a 1d6 Piercing or 1d4 Slashing (Agile) attack and an item status bonus to attack rolls, at a cost.
• Bestial Mutagen (consumable, level 1+)
  Gives you a 1d4 Slashing and 1d6 Piercing attacks.