DAVEEEE This question is awesome! 
Well im going to attempt to stay in the Robotics realm for mine
My favorite drive trains are something we call West Coast Drives in FRC or a mix of a few drives as there are different variations. The most popular and probably well known are 254’s, I have been trying to make one like theirs I really like the drive trains.
http://team254.com/blog/wp-content/uploads/2011/01/drivebase-2011-1-10-550x334.jpg
The gearboxes are also pretty freaking cool!

The oddest drive train I have ever seen will probably be a version of a kiwi drive that I saw. What makes this the craziest kiwi I have seen is that 2 wheels were 4 inches and one wheel was 2.75 inches, I have no idea what this did to help or know why the person did this. When I saw it, it didn’t make much sense. That is why its the craziest.
Most Innovative/Creative drive trains I have seen probably, would be given to the teams who design things such as 148’s Nonadrive, or 488’s Octocanum (as I call it Spamthingmicon/Baconmicon), Or 1625’s “Lobster drive” or any drive train that has 2 functions or at least 2 driving styles.
The Craziest drive train, I’ll give that to my good friend Cody with team PTD and his omni-module. Those things were just… Crazy… I took them apart, they were pretty much the most engineered thing on that entire robot and were freaking BRICKS I tell you! Maybe we will have to reconstruct one someday just to keep it for the sake of keeping it. Also I don’t want to seem like I’m being full of myself for my team Im clearly saying PTD for a reason (just to make that clear).
I do want to make a personal entry though for the worked on section.
I helped design a crab drive this year for FTC with some students I’m pretty proud of the system and I like to show it off as work of me helping students.
here are some pictures:



If you want to view the entire album (it should be public) click here
http://www.facebook.com/photo.php?fbid=10150180408805728&set=a.463736115727.249551.717065727&type=1&theater&pid=7325391&id=717065727