I did one in 1999-2000 while at school on an F1. The first six to eight months were basic product development, focus groups and testing and I didn't need to incorporate the business.
Once the proof of concept and market testing was done I started fishing for funding. 6 month before graduation signed an incubation agreement and with the help of the incubator formed an LLC. The Incubator owners became members of the LLC which helped with incorporation as well as my OPT (Optional Practical Training).
But this was 11 years ago. I am not current with the latest in immigration and student regulations. Best bet is to check it out with your ISSO (International Students) office on campus. They generally tend to be helpful.
Unless the nature of the business in congruent with the field of study and covered by an OPT -- then I would propose that this is not legal as it does not conform to the requirements of the F1 visa.
A F1 Visa is allowing you in the United States to study -- not start a business. There are Visa programs which welcome you to the United States to start a business. However, if you do not meet the requirements of the Visa programs designed for that purpose, then you are not eligible to engage in that activity in the United States.
It sucks. But that is the law.
By trying to figure out how to do a start-up (based on previous posts) while on an F1 visa you are trying to circumventing the intent of the law. You will continue to run into problems -- getting and securing your relevant business licenses for example.
Even if you don't get caught now, you could get a RFE from the USCIS 5, 10, 15 years from now -- and risk being deported.
Asking if people on the board engaged in something illegal would be like asking for marketing advice for launching a prostitution business. By answering the person may be putting their own immigration status at risk.
This doesn't mean you can't be engaged in many of the steps often associated with a start-up -- but the second you try to launch, hire employees, accept payment, receive payment . . . you know -- run or manage a business -- then you will be violating your agreement with the Government and people of the United States who are allowing you to visit their county on the premise that you are here to study and learn.