Yes, you can start many businesses with no money down, so many that it's impossible to even start listing. Most of them are services. Think of businesses like graphic arts, landscaping, gardening, carpentry, consulting, business plan writing, painting, home car, bookkeeping, so many other kinds of businesses ... what you need is customers, not a down payment. What can you do that people or businesses will pay you to do?
Creating a MAP will take no more than an hour of your time every month and will keep the lines of communication open, ensuring relationships with investors remain strong, and ultimately helping early-stage startups succeed.