I know practically nothing about cars, but when this happened with a former car (Mercury Capri) it was the catalytic converter sloughing off bits and pieces into the exhaust system, plugging it up. After a few minutes of sitting, those pieces settled out and the car would start. Once we replaced the catalytic converter, the problem did not reoccur. That probably isn't it, but it couldn't hurt to check.
The only other thing that happened was with my Subaru Impreza that had a built in security alarm that I never did activate. After several years, however, that alarm created a short or something in the works that prevented the car from starting. Once I had the security system taken off, there have been no further problems.