UPDATE - 10/9/15 : I've removed quite a few links that were either dead or led to sites that hadn't been updated for 6 months to a year. But I've also ADDED several stories to replace them. Enjoy!
The Wolff Legacy (FloorRaisin) : A 10-Gen legacy, it's been running since the game was released. Well into Gen 5 with Gen 6 soon to be on the way. Very funny with some touching story moments thrown in for flavor.
The McKee Legacy (Abby) - A 10-Gen legacy challenge by one of the Boolprop moderators. [COMPLETE!]
The Nawatt ISBI
(Kim) - One of the first ISBI challenges I've seen for the TS4. No
official rules posted yet but I imagine the basic idea remains the same.
A VERY funny read, the 'founder' Ellie is a hoot. [ON HIATUS due personal issues]
The Landgraabacy (Dashingsyndrome) - A 10-Gen Legacy with the well-known Malcolm Landgraab as a founder. Extremely funny read, and I always love legacies that star premades. Currently starting Gen 5.
CK213's TS4 Favorite Simming Moments (CK213) - Not a legacy or an actual story perse, but CK213 makes some very pretty sims and always manages to put them in the most entertaining situations. Highly skeptical at first, he's slowly but surely coming around to TS4.
The Dalton Legacy (mojot89) - A TS4 Alphabet Legacy. Hasn't been updated since March, but still a good read. [ON HIATUS]
The Lightning Alphabetacy (ZaleISBI) - A TS4 Alphabetacy with a twist; it's played on short lifespan! Hence the 'lightning' part. Very humorous, though it's also been quite some time since the last update. [ON HIATUS]
Locke Takes On the Alphabet (Ellenyarai) - A TS4 Alphabetacy, with a cute founder and a very funny read. Currently in Generation B.
The Rad Legacy (Sam) - A 10 Gen Legacy, currently starting on Generation 5.
The Belle Legacy (iheartdinosx3) - A 10 Gen Legacy, about to start Generation 4. A mix of commentary and light story-telling, a very good read.
The Bloomer Legacy (The Plumbob) - A #BuildNewcrest Legacy Challenge, just starting on Generation 5. More story-oriented, but a great read besides.
Gather Ye Rosebuds While Ye May (Rachel Rosebud) - A rotational play legacy of sorts, starts out as purely commentary and slowly develops into more of a story as it goes along. Very entertaining to watch the neighborhood grow as a whole.
Cassandra's Legacy (Audrey) - A 10 Gen legacy, just starting out but with a founder of Cassandra Goth. And I'm a sucker for those pre-made founders. Currently starting Gen 2.