Pacifist thief-play a pure thief who doesn't resemble an assassin/murderor. Zero combat.
Pure Ranger Hunter- never enters a walled settlement or a manmade structure, carry no gold, get everything you need from the wild. Hunt/camp/eat/etc.
Breton pure mage, no weapons/armour or alchemy. Loved every minute of the playthrough!
My top three are:
1. Illusionist
Major skills:
Illusion
Alteration
Restoration
Sneak
Speech
Enchanting
2. Inquisitor
Major Skills:
One handed
Restoration
Block
Alteration
Destruction
Light Armour
3. Agent
Major skills:
Marksman
Speech
Lockpicking
Sneak
One handed
Light armour
Those were the ones that I really had/have fun with. I know you usually pick 7 skills for major, but you hinder yourself if you spread your perks out too much.
To me the best builds are the ones that are the Antithesis of OverPowered. While it is difficult to make a character that is truly gimped in Skills (like putting your most used and needed skills into Misc as in games past or selecting attributes that don't help your major skills), it is possible to RP a character like this. It also helps to avoid the crafting skills to follow this route. It also helps to throw away any money you get or find and only use what is found as loot.
I love the Ranger/Explorer type builds. Mastered Archery & Sneak. One Handed & Light Armor on the side. Practice in Alchemy. Spend the Majority of your time Exploring Caves/Dungeons/Dwemer Ruins, etc. Maybe take side jobs for the Jarls and Work from Inns. Or fit the Companions into your RP and use those Radiant Quests. Use Hearthfire to build your home, but keep it as the small little Shack. A Ranger/Explorer has no need for a mansion. Get a companion of the same class as you, like Aela or Faendal.
I had a couple of builds like that. Most fun I've probably ever had in this game. I don't get as much enjoyment being a Warrior or Mage.
My most used build is a Thief type player. Light Armor, Archery, Duel-Wield, with Sneak. I also throw in a bit of magic use like Restoration and Alteration or Illusion.
I am currently having a blast playing a Heavy Armor, Sword and Board Redguard that is out to make money from dungeon delving. The shield use is another fun aspect I am sorry I did not get into earlier.
I also got very far (before the game made playing on untenable - Could not hurt Alduin in our first fight) playing a pure Illusion/Alteration/Destruction Mage.
The build that I've had the most fun with, is just a traditional Sword and Shield build (Either Light or heavy, although I prefer light) with a sprinkle of Restoration to keep the health up. I make sure not to overpower myself by avoiding smithing (Except basic Leather armor), enchanting and overpowered alchemy potions/poisons.