The Doctor's Greatest Enemy

The Doctor has three great enemies  the Daleks, of course, the Cybermen, and the one people leave out, himself. He is his own worst enemy.
Both the Daleks and the Cybermen have been enginered to be emotionless. The Doctor has his emotions-900 years of emotions. Those traits are at the core of their greatest weakness.

A Dalek

The Daleks are a race bent on exterminating all things not Dalek. They’re little ugly one-eyed octopus looking things inside a robotic casing made of Dalekanium They are not impossible to escape—its just extremely very not likely. The creator of the Daleks took out their emotions to make them better fighters.  

A Cyberman

The Cybermen, on the other hand  are not alien. Sure, they were created in an alternate universe and fell through the Cardiff rift, but they are native, so to speak, to Earth. They are humans who have been “upgraded” by having their brains put into a metal suit and having emotions neutralized, because the man who created them found emotions to be too painful.

David Tennant as The Tenth Doctor

The Doctor’s most dangerous enemy is what attracts us to the show. Not only are we watching a man fight threats beyond our imaginations, we are watching a man fight himself. The Doctor is the last of his species. Although he is an alien, he is exquisitely human. He looks like a human (or, as he puts it, humans look like Time Lords as Time Lords came first) Being a Time Lord is “a sum of knowledge, a code, a shared history, a shared suffering” (The Doctor’s Daughter). Being a Time Lord was having a connection with your people that all humans long for. (People=members of the same species, relatively generic, human=Homo sapien) The Doctor had all of that, and he blames himself for its destruction in The Time War. He had it all, and now he has nothing at all. He grasps out for a companion all the time, knowing that they can stay with him forever, but he cannot stay with them forever: at the start of the most recent revival (2005), the Doctor is already 900 years old. That is where The Curse of the Time Lords comes in--"[they] don't age. [They] regenerate. But humans decay. [Humans] wither and [humans] die. Imagine watching that happen to someone you... [Humans] can spend the rest of [their] life with me. But I can't spend the rest of mine with you. I have to live on, alone." He is truly and utterly alone in the world.
The Doctor is a mysterious man, and that is part of why we are so fixated on him. Seriously, we don't even know the man's real name! The little glimpses of his ture character, the elements of him that are carried over in all of his regenerations are little gems for us. We hang on them. We search for them. We squeeze all we can out of them.

The sixth Christopher Eccleston episode, Dalek, makes one of the first mentions of The Great Time War. We don't know much about the war, but we do know that the Doctor lead Time Lords in the fight against the Daleks, and that both races were almost completly destroyed. In Dalek, the Doctor discovers that not all the Daleks were destroyed. He finds one in a collection of alien objects in Utah owned by a misguided man named Van Statten. In this scene, Van Statten finds out that the Doctor is not human because of his knowlege about the Dalek.

DOCTOR: The metal's just battle armour. The real Dalek creature's inside.
VAN STATTEN: What does it look like?
DOCTOR: A nightmare. It's a mutation. The Dalek race was genetically engineered. Every single emotion was removed except hate.
VAN STATTEN: Genetically engineered. By whom?
DOCTOR: By a genius, Van Statten. By a man who was king of his own little world. You'd like him.
GODDARD: It's been on Earth for over fifty years. Sold at a private auction, moving from one collection to another. Why would it be a threat now?
DOCTOR: Because I'm here. How did it get to Earth? Does anyone know?
GODDARD: The records say it came from the sky like a meteorite. It fell to Earth on the Ascension Islands. Burnt in its crater for three days before anybody could get near it and all that time it was screaming. It must have gone insane.
DOCTOR: It must have fallen through time. The only survivor.
GODDARD: You talked about a war?
DOCTOR: The Time War. The final battle between my people and the Dalek race.
VAN STATTEN: But you survived, too.
DOCTOR: Not by choice.
VAN STATTEN: This means that the Dalek isn't the only alien on Earth. Doctor, there's you. The only one of your kind in existence.

"Not by choice." That is the core of this Doctor. The Doctor has severe emotional trauma, and the results of that are very pronounced in the Eccleston interpretation. He is a very sad, depressed man. This is where his desire to die (caution, spoilers) starts.

In the 2011 episode "The God Complex", the Doctor (Matt Smith) sees his greatest fear. Although it is never explicitly stated what is in the Doctor's room, some major hints get dropped. His room's number is 11 and he is in his eleventh incarnation. When he opens the door, we hear the TARDIS's cloister bell, which means a red alert. When he sees his greatest fear, he says "Of course. Who else." One does not simple say the word "who" in the series without it being a pun. It is all too clear.
In a way, we are all our greatest enemy. The Doctor may not be human, but he has so much humanity.