Here is a list of challenges. Most of these are just a jumping off point to give some new life into the game, either as a new way to start or a jumping off point for some roleplaying.
The Slave Escape:
I first saw this idea from Danae, who made a mod specifically of this, but there's no need for the mod.
Simply, you make a new character, recommended Argonian or Khajiit. Travel to the Dren Plantation, drop all items/money outside and take up the role of a slave. You may not leave the plantation grounds until you've either killed Orvas Dren or freed all the slaves.
I 'did' this, up until it wasn't fun any more - just constantly reloading saves until I could knock someone's fatigue out before they murdered me. It was interesting to sneak around and gather supplies, slowly equipping myself into a ready adventurer, but the challenge itself is a bit of a drag.
Bloodmoon at Level 1:
I saw this idea first from MickyD's youtube channel. No mod necessary, just take boats up to Khuul and ship off to Solstheim.
Although it is completely possible to start the Bloodmoon quests at level 1, the lack of resources on Solstheim is where the challenge actually begins. There are very few potions available, basically no spells, and equipment is all tied to quests. The main quests spike in difficulty pretty early, but the Ravenrock quests are reasonably accomplished. The biggest issue is the Frostmoth Smugglers quest. This is a long dungeon filled with paralyzing weapons, with your limited heals and no paralysis potions available. Even leveling up a few times before getting in there doesn't make it easy. The last few quests are tough, but being a werewolf can shore up most of the challenges.
I 'did' this one as well, doing most of the Raven Rock quests (Razing the Forests was finished but by then, I hated the experience so much, I went back to the mainland) and up until the Smugglers quest in the main storyline. It was fun and exciting, I went on with that character to then do the Imperial Legion and Imperial Cult quests before finishing the main story. I had a lot of fun with that character.
Tribunal at Level 1:
Basically the same story, but way, way harder. I tried this for a bit, but constantly was using the Orc racial ability, either dying or killing a single enemy, then waiting for 24 hours to do it again.
Any Island/Place at Level 1:
So this is sort of the same premise, but you just head for any island and do as much as you can there before you can leave. I had a lot of fun with the Province Cyrodiil mod, as well as wandering around Skyrim: Home of the Nords. Tamriel Rebuilt in general has plenty of places for you to go, or you can stick to Dagon Fel and the surrounding area. This is really just a way to give a character a new intro to the game. Gather as much wealth as you can from a town and the surrounding ruins before running up against an unbeatable obstacle. You can do the same by staying inside a city (Ald'ruhn has walls to help, even), but there's little to do in cities apart from 2 or 3 quests.
The exception to this is a City Folk run. You play entirely in the cantons of Vivec - there's about 10-15 quests all accomplished there, plenty of services and your pick of where to set up as a home.
The Full Random:
There are mods that give you a random spawn location, or you can just roll on tables for your character's race, pre-made class, and birthsign. I think this is most fun with an ironman playthrough, but as soon as you get to a town, it basically no longer matters - as you can just travel where ever is safest and build up from there. You can also use something like
this to make your character. I think if I were to do this again, I'd do it Ironman and also not look at the map after being randomly spawned in the world.
Smaller Challenges/Just Things To Do:
The Seven Graces:
Travel the Seven Graces by foot, ending in the holy city of Vivec. I just did this and it was absolutely wonderful. I gathered herbs along the way and sold them in the temples, then burned through most of that money on equipment from enchanters and at the shrines. It was actually really fun. From Balmora Temple - Kummu, Ghostgate, up past Caldera to Gnisis, Koal Cave, all the way down the West Gash/Bitter Coast to Vivec's 3 shrines. You could do the same for the 4 corners later in the temple quests.
Why Ride When You Can Walk?:
This is a pretty common game-rule. You have to walk to a place first before you can use fast travel to get there. I guess Mournhold is an exception, unless you've got Tamriel Rebuilt.
Service Refusal Run:
I think this is interesting, if not entirely exhausting. The idea is to never do a service if you can do it yourself. Don't repair stuff, buy hammers and do it yourself. Potions? More like you buy the ingredients and make them yourself. Enchantment, too, if you're nasty. Travel too, if you're a masochist.
N'Wah Challenge:
Set all attributes and skills to 1, then see how long it takes to finish the main quest. Apparently this is just a very slow start, but, because of how leveling works in Morrowind, leads to you being the strongest possible character by the end, which is neat.
If I find/remember/think of more, I'll update this post.