It's nice to see how other folks have developed their own personal toybox storylines. I myself have several, mostly for other LEGO community sites, that I'm very fond of. But my "main story" has been going on in my head (although I have written some of it down/taken pictures/created comics and the like) since I was a kid. I never had a great deal of access to LEGO as a child, as it was too expensive. During my first years of college, however, I went bonkers and bought a great deal of it. A little while later, the fantasy Castle stuff came out and I went even crazier (Castle was one of the things I wanted desperately as a kid...the fact that it was creative fantasy pushed me over the edge).
Anyway, my original story wasn't really a story at all. Just primordial chaos and toybox wars. As I got older and fonder of storytelling, I began to develop more interesting and persistent characters. One of my first was a technomancer who developed the secret to artificial intelligence. It seems like a small thing, but the one thing I've always liked about that era of my childhood play experience was that the technomancer didn't create an army of robots to subjugate the other minifigs but, rather, built all of his AIs into vehicles. He traveled around like an adventure hero and diplomat combined, speaking with the other figs, saving them from dangerous situations, and teaching them purpose beyond endless fighting. It didn't work like he'd hoped, exactly. The various minifigures began to group into factions with strong, central ideas and goals. Then, those factions began warring with one another. Of course, it wasn't always physical conflict. I suppose it was about that time in real life that I first saw reruns of Mission: Impossible on a television channel called FX. I'm sure that influenced my storytelling leanings somewhat for years.
I have a family of my own now and I'm pleased to note that my oldest son has his own developing storylines. I still buy LEGO from time to time, when I can afford it. I reuse some of the figures for various community based stories but my main story uses all of them. Currently, they remain grouped into factions, though those factions have grown somewhat over time. I'm a huge fan of multigenre, so the individual factions (for the most part) are mashed together.