Well--I'm not sure, tsuwm, if you meant an opposite match to the term in which instance full- or all-brainer would work, I guess, though I don't like either of them very well, or if you meant a word for an opposite meaning (doubtful) in which case poser, conundrum, etc. would do.

Let's see--no-brainer means no (or little, as per the official def.) thought is required. So the opposite would need a word for full amount or maximum thought. ... ! Brain-buster?