(I'm most familiar with the modern Doctors, Nine thru Eleven. Saw some of the Fourth Doctor episodes back when they were showing on PBS in America, although I don't think I ever managed to catch a complete story arc. And snippets of most of the other Doctors over the years. So the following suggestions are more in the general line of attitude, charisma and the actor in question showing a previous capability for playing odd or eccentric characters, rather than matching a particular actor with a particular Doctor.)
For some of the earlier Doctors:
dealt with an alien invasion!
Now some suggestions for the more modern era:
Feel free to add your own suggestions in comments.