It seems likely in my experience that starting a legitimate small business (that has a real chance of success) with absolutely no money is not very common. There are always going to be some basic start-up expenses in marketing for example like creating business cards, flyers, or building a web site. However there are a few types of small businesses that can be started up very cheaply, maybe with as little as a few hundred dollars initially.
In my opinion the best and most inexpensive type of small business to start-up today would be a service type business that can be operated from a small home office which will significantly help reduce your overhead expenses. A good example would be a residential or commercial cleaning business which does not require a large investment in equipment, can be started out owner operated with just a few accounts and can easily to operated from a home office initially. Other top examples of cheap businesses to start could include a handyman or home services business, a pet sitting service, a senior companion service, or some type of computer service or repair business.