This is not the best idea in my opinion, and let me explain why.
The chance for spam bios would increase and they could just have never-ending bios that go down forever, and not to mention sometimes it's nicer to have much shorter, concise bios. We don't always need to know what your character's breath smells like in the morning, or about that one time they went for a stroll in the park.
Although a much larger space for bios would be nice, unlimited would be pushing it, and I don't think the game could handle that for long.