One thing I had fun doing, especially in The Sims 3, was do reverse-storytelling. Usually works best with amnesia stories.
Basically, you start off with nothing, no home, no supplies, no money, and pick and choose when to add to the story. Mostly, you'll be using your sim's perks and flaws to your advantage to keep them alive, and make use of a few cues planned or not (random events and such) to add another chapter to the story, before or after the amnesia zero-point (where the story began/continues from etc.). You can always justify cheating by adding funds to buy a fancier house to add to the storytelling as if they remembered where they lived before and so on.
I had a story evolve from a John/Jane Doe having forgotten their ID (and also start with pretty much nothing else) to making use of the local gym and park as a de-facto home with some stuff they had to mooch off the good folks of the town (starting traits was that of what would naturally come to mind after losing it), mooching and robbing random sims, to doing some menial labor, and eventually becoming a decent politician (working with aspiritions), and then a new family 'moved in', turns out to be a rather troperiffic evil mafia group of sorts, and as a twist, the main character worked with them before. They got captured, and were living in a cage in the lair, temporary control given to the bad guys to develop them a bit and sustain their prisoner (the amnesiac gone mayor by then), after recovering the memory of working with them, they get a change of heart from working with them again, and took the opportunity to act like one of them again, and slowly killed them off one by one. By the end, there was a choice: Become Leader of the Free World, or take the place as the Ultimate Evil Villain... and Leader of the Free World? The end is a mystery.
In a sense, I had the idea to do something like this for a Sims 3 LP; but as I was test-flying the idea, I realized how much work would actually go into it, and due my laziness, didn't go through with it. Nonetheless, I had fun with it. Feel free to use this concept for your own Sims 3 LP.