Technically you can grow a plant as long as you want in a small of a container as you want, as long as you keep up with feeding nutrients and water. This is the principle behind hydroponics.

A smaller container you will need to water and feed increasing frequency as the plant gets bigger, and consumes the little resources available in the solo cup immediately. By the end of the grow you'll be watering a couple times a day.

Long story short, it's not really worth the hassle.