The first two stories, An Unearthly Child and The Daleks; then I think the Dalek Invasion of Earth, and finally The Tenth Planet (where unfortunately most of the crucial fourth episode is lost). The Daleks Master Plan is also really good but unfortunately only three of the twelve (or thirteen, depending how you count) episodes survive on video.
Second Doctor
Sadly most of the stories have been destroyed, at least in video format; the best surviving ones are the Tomb of the Cybermen, the Mind Robber, The Invasion and The War Games.
Third Doctor
Starts really well with Spearhead From Space, and then Inferno; then try Terror of the Autons (first story with Jo and the Master), The Three Doctors (a bit silly but if you have followed the course so far you will be keen for nostalgia), Carnival of Monsters and The Green Death
Fourth Doctor
Well, you’ve got two of the best – some would even say the two best – already. I would also recommend The Ark in Space, Pyramids of Mars, The Robots of Death, The Talons of Weng-Chiang and City of Death.
Fifth Doctor
Much more difficult to pick here. The only ones I have re-watched recently are Earthshock (not so great), The Five Doctors (OK) and The Caves of Androzani (brilliant – directed by Graeme Harper who has also directed several of the better episodes of New Who). But I also plan to go back and re-watch the linked stories Kinda and Snakedance some time fairly soon, and I have very good memories of Mawdryn Undead.
Sixth Doctor
There is a widespread view in fandom that holds that all the Sixth Doctor stories were terrible. I agree with that view, but you may want to make up your own mind.
Seventh Doctor
I really liked two of the last three stories ever broadcast, Ghost Light and The Curse of Fenric. I don’t like Remembrance of the Daleks, though almost everyone else does.
Eighth Doctor
Only one story, but you have to have watched it to gain your fannish credentials!
Discussion welcome!
It’s clear you’re not the intended audience for Girl Genius; in future years you should probably spare yourself the effort, because it’s not going to change its essential character.
That said, it’s almost certainly going to win again this year, because it IS hitting its very large target audience very well, and that target audience clearly has a large Hugo following.
If I buy a membership for next year, though, I’ll be nominating Namesake in that category. This comic gets more amazing every week.