The Doctor has beaten a lot of powerful enemies against long odds. But who do you think he would lose to, or at least be far from certain of winning?
My pick: President Palpatine/Darth Sideous. It starts out as a classic Who adventure: The Doctor finds a formerly peaceful Republic torn asunder by war. As the Doctor investigates, he finds that something rotten is going on: a mysterious evil known as Darth Sideous is behind it all. The Doctor finally unmasks President Palpatine as Sideous and then… well, IMHO nine falls out of ten the Doctor dies or has to retreat in total defeat. Short of a very lucky break, a averagely-gifted psionic with an outstanding knowledge of advanced technology simply isn’t going to fare very well against a Sith Lord.
But on the other hand, measured in pure fighting toughness, the Doctor should have been totally destroyed long ago. Hasn’t the Doctor faced enemies before who could squash him like a bug, but somehow prevailed? Is the Doctor’s real gift gigatons of sheer dumb luck? He has a genius for finding the one tiny thread to tug that can bring down even the baddest enemies. He seems to literally always be in the right place at the right time. The hand of Rassilon perhaps?