Ask HN: Can you get a job at Google without a degree? 14 years ago
You might focus on trying to get any job at such a company, and then be really proactive about showing off your designer/developer skills. I knew an Apple retail employee without any college education who landed an engineering job in Cupertino through an internal talent program.
You might also consider looking at startups or incubators. They tend to be less interested in credentials and more interested in ability to execute than some of the larger companies.
So, it's possible, but I would agree with the other poster that trying to stand out among a normal application pool would be really difficult.