It's been nearly a decade since this was asked, and by now it's hard to find an installation of Skyrim without at least the DLCs. There's no guarantee that Khajit are the strongest unarmed race, and Fists of Fury overloads unarmed combat on its own. On unmodded Skyrim SE with all content installed (DLC and CC), what's the maximum damage available to a player of any race without exploits?
Here's some preliminary work:
Khajit, Argonians, and Vampire Lords have 10 base damage. Khajit also come with +12 damage from Claws. Vampire Lords have leveled damage from Vampire Claws (UESP) up to +40.
The Vampire Lord skill tree also mentions Poison Talons doing 20 pts of poison damage to melee attacks.
Necromage from the Restoration skill tree apparently affects Vampire Claws (UESP), increasing the effect by 25% (to +50)
Fists of Steel adds damage based on the base armor value of the equipped gauntlets.
Without the potion exploit, the maximum we can enchant Fortify Unarmed to is +10, at enchanting 100 and a grand soul. The Brawler's Daedric Gauntlets start at +18, and the Fists of Randagulf are +20. However, the Fists of Steel modifier favors the BDG on net, as they have a higher defense bonus.
The Ring of the Beast from Dawnguard provides an additional +20 damage.
Anyways, I think if I'm doing the math right:
10 | Vampire lord base damage
40*1.25 = 50 | Vampire Claws plus Necromage
20 | Poison Talons
18 | Brawler's Daedric Gauntlets
18 | Fists of Steel
20 | Ring of the Beast
Addendum: I recently played with fortify unarmed on an alchemy-exploit game and got the fortify unarmed enchantment well into the ludicrous range. It's easily able to one-shot dragons, and significantly easier to pull to that level than smithed items.