Wow, tons of great suggestions here. Thanks, everyone. MMM, I especially like the idea of keeping track of a user's scores and making the questions increasingly harder based on that.
The reason why most of these trivia apps are multiple choice and tap-the-right-option is because that's the far more accurate way to build the app in terms of being able to check answers. It's fairly easy to tell the app that "Option C" is correct, whereas if the question is a type-your-own-answer deal, it's harder to see that the user really meant "George Harrison" when they flubbed up and typed "Geroge ...