I have a really crazy drivetrain idea. Instead of 4 wheels, put 6 wheels on a robot, and make the center one lower! That way, robots will have a shorter wheelbase for turning but high traction all around!
Of course I'm kidding, but I've always wondered why people take the "let's do something weird for the sake of it" approach to prototyping, rather than engineering a drivetrain that tackles a design aspect differently. I think you should totally build a crazy drivetran, don't get me wrong - but maybe it would be a better exercise to figure out what current drivetrains you use don't do very well, and then figuring out a new drivetrain optimized toward different design criteria than the bog standard.
The latest innovative drivetrain in FIRST (nonadrive) probably wasn't thought of by going "hey dude, what if we did THAT?". I imagine they specifically looked at a way to accomplish something not present in a 6wd (strafing) without some of the drawbacks of drivetrains designed for that (swerve, mecanum). Maybe you can do the same thing - and come up with something incredibly crazy, but useful too!