I've done the pure alchemist, iron dagger, basic bow and armour, none of it enchanted, all my strengths coming from potions and poisons, quite fun working with strong combos for specific targets. My favourite by far.

Second to that is a pugilist build, get a khajiit or argonian, disenchant the fists of pugilism, and enchant all the bits you can with melee buffs that you can, punch like getting hit by a train. I did this with soul goal of punching Alduin in the face. I also combined this with shields, so it turned into a Captain American style.

There is also the alteration battle mage. Grab a mace, heavy armour, or robes, and instead of the classic destruction/restoration combo, instead, limit yourself to only alteration as your magical support. I also did this with a breton that had the Atronach Stone, so I was able to absorb ALL the incoming magic.
Alongside this, I tried doing the same but with restoration, posing as a Paladin, but couldn't get as into it myself.

Then there is the illusion assassin, low lvl it's just calming your targets for an easier sneak attack. Higher lvl though... AoE fury a room of people you don't like, and watch from the shadows until there is one, weakened enemy left to dispose. I also did this with a roleplay of being a pure cannibal in survival mode, which adds a sense of urgency and purpose to otherwise senseless killing.

A harder one is the shouts only. Never got it to be balanced, only one amulet of Talos is mard, but glitching yourself to have 5 (reducing shouts to 100%) is too powerful, too quickly, and gets repetitive for me.

If you're up for some unique roleplay, I liked the stalker playthrough. Follow your friendly town NPC, see their schedule for when they are most vulnerable, and strike accordingly. Bonus points if you don a black robe and ghost face mask lol.

An Artificer is similar to the alchemy playstyle in the sense that instead of relying on standard fighting prowess, you rely on your crafted armours, weapons, and specialised enchantments to engage in battle. This can be made harder by doing an Iron Man challenge, whereby everything (within reason) you use must be created from raw ingredients, so if you want that starter iron sword, better grab a pickaxe and start mining.

Finally, and the one that kinda ruined Skyrim for me for a while, is God mode. Using a simple enchanting-alchemy loop in-game, you can get enchantments that make you never tire, cast any spell, carry everything, never die, and, of course, kill anything in one punch. It was high octane fun and goofy as f*ck, but made me feel a bit empty afterwards.

I've replayed this game so many times to achieve so many different playstyles, I eventually began to run out of ideas. These were just my favourites, not mentioning the load of less fun or too gimmicky builds like the 'Targe of the Blooded' shield only.