Well, let's see...
There were several monotype challenges that were started, but never finished. Thus, there are only five types that nobody ever tried, but there are also only five types that anyone ever completed (bear in mind I'm strictly talking about Monotypes recorded in the challenge thread.)
Types that haven't been tried are Dark, Dragon, Fighting, Flying and Ghost. Dark is problematic for the reasons listed above, Ghost is problematic because there are only two families of Ghost-types, and Dragon is problematic for both reasons...however, upon reading through the challenge thread I discovered that using Game Shark is actually allowed in GSC MTCs, just for getting monotype battlers at the start of the game. So you could Shark up a Dark-type or two, or a Misdreavus or a Dratini. On the other hand, Flying would be easy to get started with Pidgey and Hoothoot both in Route 29. Fighting...might be a problem, but again, you can Game Shark a Machop or something.
The only five types that were ever actually cleared are Fire, Ice, Normal, Steel and Water. It should be noted that the Ice MTC was started by Game Sharking in Seel and Swinub, and the Fire challenge was cleared with the help of Ho-Oh before the thread host specified that Legends weren't allowed.