Sex, entertainment, games, movies, health, fitness, education. It's takes a little twisting but you can frame every one of these such that they are in fact problems that need solving.

Yeah it's easy to play word games to describe everything as solution to a problem but that's really just cheating to be able to actually say "every business is the solution to a problem".

For example, "entertainment/games are the solution to boredom". You'd only say that if you really wanted to back the idea that every business must be the solution to a problem.

Hard to argue that Twitter solves a problem, nevertheless there's an awful lot of people using it constantly. In the "every business must solve a problem" world, you could say "Twitter solves the problem of people needing to communicate with all the other people in the world in public conversations", but that's really trying to force it.