Chameleon arch: Okay, who didn't know that I love the concept of the chameleon arch? Come on. That one's obvious. "Human Nature/Family of Blood" is my favorite episode, and the concept of the arched human having to come to terms with actually being someone else is so compelling to me, I've created a whole AU around it. I haven't written anything recently in that vein, but I've always wondered how it would go if I wrote a story involving a different Doctor being arched, not just Ten as usual.
Multi-Doctor encounters: When I first got into DW, I loved finding stories (both canon and fanfic) in which the Doctor encountered himself - the interactions are just stunning. I think the best multi-Doctor story ever is "The Light at the End", a gorgeous anniversary special that spotlights each Doctor equally and plays to each of their strengths, on top of a great story. But I have to give an honorable mention to "Peri and the Piscon Paradox" for the absolute best (and hilarious) example of the Doctor fighting with himself.
The Doctor assisting himself via time tricks: I'm talking instances where the Doctor either sends himself messages from the future or the Doctor returning in a later incarnation to finish what he started. An example of the former is "Battlefield", in which the Doctor encounters messages left by him later (including a stone carving the archaeologists couldn't decipher but the Doctor reads "Dig here", because it was written in his own handwriting). I can't think of a canon example of the latter, but I'm sure Big Finish has done it a couple of times. This kind of thing is rare, because it really is a plot cheat and when overused, destroys the story and the show.
Superpowers and magic: I just love superpowers, which of course is why my only true AU (meaning, not actually in a fandom universe) is a superhero world. Magic is also a big draw for me, though I prefer my magic structured, with rules and consequences, like Sanderson's Mistborn or The Dresden Files, and not like Harry Potter (though I still love HP). Most of the things I've ever been a fan of have either been straight science fiction or had powers or magic of some kind. When I was a kid up through college, those were tokusatsu shows, "The Greatest American Hero", "Forever Knight", and Spider-Man and X-Men comics, to name a few. More recently, there's DW itself, HP, the MCU... Heroes was just amazing - for the first season. Not so much later.
Psionics: It's kind of a subset of superpowers, but I love stories with telepathy and other psychic powers. The Doctor helping Donna to hear the Ood song was just perfect for me.
Wings: I love love love wings. I don't know why. Part of the reason I am ITCHING for Good Omens this year is because David Tennant will have wings. And black ones! Which my Tenth Doctor character in Guild Wars 2 has been wearing for years. Because wings on David Tennant. And because if any Doctor would have black wings, it would be Ten. But still, David Tennant with wings. *cough* Okay. Back to the trope. This is a weird one because while I love wings, I couldn't write wingfic. "The Doctor, but with wings!" or whatever just doesn't make a good story for me, even if it's completely AU. Having wings isn't something to base a story on. It's like basing a story on having feet. Okay, I'm probably just rambling now.
Glowing eyes: "42" is one of my favorite episodes, and it totally helped that the Doctor had glowing eyes in it. Well, to be perfectly honest, burning eyes. How in the world was he not blind after that?
And lastly, I want to give a nod to the doppleganger crossover. I'm not so big into it now, but back when I started writing, I really enjoyed crossing the Tenth Doctor with other DT characters. I've always wanted to do some with other Doctors, but unfortunately I know nothing about any of the other actors' works.