Success in a business context means a positive rate of return on the resources invested. Or to refine one step further, a return that is positive and superior to the next best alternative for the deployment of said resources at a given level of risk.
See Amazon (or as noted below, Instagram).
However I also think that is a problem for many people who want to startup. They don't usually think about how they'll show a profit and manage to run their business within that profit. Investor money gets them started and an exit by selling an unprofitable yet growing business to a larger company is the goal.
Instagram, among many others, most certainly generated ample return on its initial invested resources without ever generating profits.