Why doesn't the Doctor consider himself a hero?

When the Doctor saves planets, people call him a hero, but he replies by saying he's not a hero. Like this one time:

I'm not a hero. I really am just a mad man in a box

and this other time:

ROBIN:You are her hero,

DOCTOR: I'm not a hero.

Why doesn't the Doctor consider himself a hero? He is: he saves planets and defeats monsters.

Answer

Warning: May contain spoilers. (Mainly from the rebooted series.)

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From an external perspective he is typically considered to be either a hero or anti-hero depending on the incarnation and which acts one is considering.

The Doctor himself considers himself to not be a hero for many reasons.

Firstly because he (almost) never seeks situations which would put him in a position to perform heroic acts. Contrast this to 'traditional' heroes who often make it their job to be heroic, for example warriors such as Heracles and Achilles, or more 'everyday heroes' such as firefighters, police and (ironically) doctors. Instead, the Doctor simply travels and trouble usually finds him.

During his travels, the Doctor committed a great many crimes and morally questionable acts, many of which we still don't know the full extent of (e.g. the events of the time war). Aside from the numerous murders he's committed, he's been a guilty party in a number of genocides:

• The Tenth Doctor went back on his fourth incarnation's 'have I the right?' speech and outright murdered the entirety of the Racnoss empress' children, thus making him accessory to genocide (if not the main perpetrator) when the British army wiped out the empress herself. (The Runaway Bride)


• The Fourth Doctor destroyed Sutekh, the last known Osiran. (The Pyramids Of Mars)


• With Leela's help, the Fourth Doctor destroyed the last of the Fendahl. He dropped the remains of the Fendahl into a supernova to ensure their permanent destruction. (Image Of The Fendahl)


• The Fourth Doctor killed the last of the Great Vampires and the only vampire survivor of the Vampire-Time Lord war. (State Of Decay)



• The Sixth Doctor wiped out the newly created Vervoid race on Hyperion III. A crime for which he was actually tried in court. (Terror Of The Vervoids)


• The Eleventh Doctor was responsible for wiping out the last of the Saturnyns. The last one to die stated that she hoped the species' extinction would haunt his conscience. (The Vampires Of Venice)

Some miscellaneous examples of the many (albeit less serious) crimes has committed over the years:

• The First Doctor stole the Tardis he now travels in (it is still stolen property, the Doctor never returned it or obtained lawful ownership).


• The First Doctor kidnapped Ian Chesterton and Barbara Wright. (An Unearthly Child)


• The Eleventh Doctor was prepared to lobotomise the last star whale in order to save a single ship of humans (a much overpopulated species). (The Beast Below)


• The Eleventh Doctor stole the Byzantium's Home Box from the Delirium Archive. (The Time Of Angels)


• The Twelfth Doctor has broken into the bank of Karabraxos and stolen a memory chip and a gene suppressant (and arguably a slave if one considers a slave to be the property of a person). (Time Heist)

On top of which he certainly has a mean streak.

• The First Doctor was perfectly happy to deceive his captive audience and his granddaughter for the sake of investigating the Dalek city on Skaro, endangering them all in the process. (The Daleks)


• The twelfth doctor has forced his companion to coax humanity into deciding whether or not to kill an unborn creature to save themselves, all whilst knowing full-well that choosing to not kill it would cause no harm to the earth. (Kill The Moon)


• The Twelfth Doctor has also had various violent arguments with Danny Pink over time, in which he hammered home his prejudice against soldiers.


• The 'Timelord Victorious'. Enough Said. (The Waters Of Mars)

The Doctor remembers the times he's done wrong and the times he's saved lives, and he knows that he doesn't do the things he does to be heroic. He's just a time traveller trying to help out where he can, and he doesn't always get it right.

I dare say if he had the choice he would rather the majority of his trips featured as little peril as possible, like the space tourist he longs to be. Being able to visit a nice hotel without its architect going mad and its inhabitants turning on each other; being able to take a ride on the last ship out of earth without ending up in a whale's mouth; being able to take a trip to a planet made of diamond without encountering a strange mimicking entity; and being able to go to a space theme park without it being overrun by cybermen. Like how Sherlock Holmes retired and took up bee keeping or how Winston Churchill loved to tend to his garden. This was very much alluded to when the Doctor became Victorian teacher John Smith, before being forced to give it up to resume his true persona since he was the only one with the knowledge to save everyone. (The Family Of Blood)

Finally, he's hardly first person to be heroic and then vehemently protest that they are not a hero. Many veterans of various wars have refused to accept their medals because they believed they were just doing their duty, no matter how important what they achieved was or how deserved the accolade might be. There are also various literary characters who have similar feelings about their perceived heroism.