Actually, Facebook, Twitter and Farmville are great examples of products nobody said they wanted and didn't solve any perceived problems, but are succesful any way.
I've compulsively weaned myself off the "solve a problem" mentality because it gets in the way of building truly innovative products, IMO.
You're right. You don't have to solve a problem. But you do have to create something people want, or something that appeals to them. Technically, there aren't many "problems" in life besides survival. Everything else though has to appeal to something in the Maslov hierarchy
I've compulsively weaned myself off the "solve a problem" mentality because it gets in the way of building truly innovative products, IMO.