Boy, whoever came up with this list didn't make it easy, did they?
I'm the kind of person that loves everybody and everything (except the bad guys. THEY'RE THE BAD GUYS - YOU'RE NOT SUPPOSED TO LIKE THE BAD GUYS! *ahem* That's for another day). It's really hard for me to choose my favorite incarnation of the Doctor. But if I have to choose - right this very minute - it would have to be Ten. And probably not for the reasons you would think (and it's kind of a long story, so just go with me here).
Doctor Who - David Tennant by *thecosmicdancer on deviantART
From my days in the Harry Potter fandom, I knew David Tennant (who played Barty Crouch Jr. in the Goblet of Fire movie) had been cast as the lead in some British sci-fi show that sounded interesting, but I didn't have time for it back then. Fast forward a few years later - all the Harry Potter books have been published, the movies really aren't that good (sorry - they're not), "24" and "Heroes" have both been canceled, "CSI" has gone off the rails. I'm in school, but I don't have a job and my family life is just going absolutely bonkers - I just want something to get into that isn't one of those massive "Responsible Adult" type things - I have to have something that's a bit frivolous and just plain fun. A few of my online friends were fans of "Doctor Who," so I figured I would watch a few episodes to see if I liked it.
I found "Rose" online and I quite enjoyed it, even though the Doctor in that one wasn't David Tennant. At the time, my brain didn't put together the fact that Christopher Eccleston had been in "Heroes," so I was a mite confused as to who this guy was. Happily, Wikipedia cleared that up in a hurry and subsequently blew my mind when I learned this show had been going on more or less since 1963! (What the - ? O_O!). Then I remembered that we had BBC America on our satellite package and I checked to see if they had "Doctor Who" (looking back now, I realize that's kind of a stupid question). They did, so I set my DVR to record everything they had.
The first episode I saw was "The Shakespeare Code" and I absolutely fell in love with the Doctor from that portrayal (Harry Potter references - FTW!). The next day, BBCA showed "Gridlock" and just from the Doctor's description of Gallifrey and the Time War and the utter heartbreak and devastation in his voice and on his face - and this after all his hilarious antics of jumping on the tops of cars and showing off for Martha - I knew this show was something special. I wanted to keep watching and I'm very happy I did (I went back to "The End of the World" and watched everything in order. I learned that Nine is very awesome too). I was sad when Ten regenerated, but I still knew he's the same Doctor, no matter his face or his personality (or even his clothes).
Long story short - Ten was the first incarnation of the Doctor that I really got to know and love and he's the one that kept me coming back for more. So, no matter how the Doctor regenerates in the future or which of the Classic Doctors I get to know (or which of those become my favorites as well), just for getting me into this show, Ten will always be my Doctor.