as may be obvious, anarchists certainly have started businesses, including businesses that have been (at various times) quite profitable. AK press leaps to mind as a current example (although profitability is in question for any book seller in this day and age).
but scratching at the question a little leads to further questions - what do you mean by profit? profit to make a single person wealthy? profit to support the workers of the business? profit to continue doing the thing that the business does?
you could ask what charity means along similar lines to the above. the people who run charities frequently support themselves as part of the process. so, that kind of charity? or the kind of charity of a small business that just makes enough money to continue existing, without benefiting the owners/workers monetarily?
and whether anarchists start these things is different from the business or charity being anarchist as projects. my argument is that anything that does more than supports itself is not anarchist, but that's just the best position i've come up with so far.
the anarchist line is that we are against capitalism, and that we are against charity (as distinct from mutual aid). and then there is living in a capitalist world, which means that most/all of us have to do things that we don't want to do.