The Doctor's memories
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The Doctor's memories were a core aspect of their identity. When companions left, memories were what remained for the Doctor to hold onto. (AUDIO: Relative Dimensions)
Though nothing could be more important to seeing the time traveller through, for stringing their many lives together, amnesia was a common problem for the Doctor. On several occasions, most often with post-regenerative amnesia, the Doctor could not even remember their own name, and had to find an anchor which would allow them to restore who they'd been before. (TV: Doctor Who, PROSE: The Eight Doctors, The Gallifrey Chronicles, AUDIO: Something Inside, Orbis, TV: The Woman Who Fell to Earth)
On some occasions, memories were actively suppressed from the Doctor. By one account, the Division removed all the Doctor's memories of an unknown number of lives they'd spent as the Timeless Child. Afterwards, the Doctor became a child again, and the First Doctor began a new life on Gallifrey. (TV: The Timeless Children) These earliest memories were stored in a biodata module, kept aboard the Division's outpost in the Void. (TV: Survivors of the Flux)
Crossing their own time stream
When multiple incarnations of the Doctor experienced the same event, the impact on the Doctor's memory varied. In many cases, knowledge of their future was lost, either as a natural result of their time streams getting back in sync, (TV: The Day of the Doctor) or as a result of their actions during the adventure. (AUDIO: Regeneration Impossible)
However, as with the Eighth Doctor briefly recalling all the times he'd met River Song, (AUDIO: Songs of Love) the Doctor was often able to "catch up" as the event was happening. The Tenth Doctor could finally remember the Fourth Doctor's activities seconds after they happened. Though he could not predict anything, he was able to track his past self, just by memory. (AUDIO: Out of Time) The Fifth Doctor also managed this, watching over his first three selves by memory during the Game of Rassilon. (TV: The Five Doctors, PROSE: The Eight Doctors) The Eleventh Doctor could "almost remember" as he joined events, but lacked many crucial details. (TV: The Day of the Doctor) Critically, the Tenth Doctor was able to depend on the Fifth Doctor's memories of his own actions to know how to resolve an issue with the TARDIS when it collided with itself. (TV: Time Crash) Later, moments into bumping into each other, the two Doctors remembered each other. (AUDIO: The Gates of Hell) Finally, during an encounter between the Fifth Doctor and the Seventh Doctor, both mentioned multi-Doctor events they could remember, including their involvement in the salvation of Gallifrey (AUDIO: Cold Fusion) by the Tenth, Eleventh and War Doctors. (TV: The Day of the Doctor)
History
Second Doctor
The Time Lords erased the Second Doctor's memories several times whilst he was working for them following his trial. (PROSE: Save Yourself) They made him forget his companion Tarlos. (AUDIO: Colony of Fear)
Third Doctor
The Time Lords removed some of the Third Doctor's memories, specifically those which would allow him to operate his TARDIS during the Doctor's exile on Earth, following the Second Doctor's trial on Gallifrey. (TV: The War Games, Spearhead from Space) These were eventually restored, after the Third Doctor helped defeat Omega on a mission for the Time Lords, with the help of his first and second incarnations. (TV: The Three Doctors)
Sixth Doctor
The Sixth Doctor's memories were affected by him being taken out of time to Space Station Zenobia, leaving him unaware of the circumstances of his and Peri Brown's separation. (TV: The Mysterious Planet, Mindwarp)
Mortally wounded by Zor, the Doctor was found and saved by Jack Harkness, who he learnt was a former companion of a future Doctor. Preparing to enter a healing coma to fully recover, the Doctor assured Jack that he would "edit" the memory of meeting him prematurely. (AUDIO: Piece of Mind)
Seventh Doctor
The Seventh Doctor edited some of his "more useless" memories. (PROSE: Timewyrm: Genesys)
Eighth Doctor
The Eighth Doctor in particular was prone to amnesia. Immediately after regeneration, he lost all memories and forgot who he was. Interacting with the world around him, the Doctor was able to remember his clocks aboard the TARDIS, being with Puccini before he died, an encounter with Leonardo da Vinci, an early memory with his father on Gallifrey, and finally, his own identity. (TV: Doctor Who) Later on, he lost all his memories again, in a trap set by the Master. By mentally linking with his previous seven selves, the Eighth Doctor was able to regain what he'd lost, one incarnation at a time. (PROSE: The Eight Doctors) On another occasion, he erased his own memories after being forced to destroy Gallifrey, in order to store the compressed Matrix within his subconscious. (PROSE: The Gallifrey Chronicles)
The Eighth Doctor also lost his memories inside the Cube, designed to suppress the psychic abilities of decommissioned soldiers, making him a useless subject for interrogation, to Eric Rawden's immense displeasure. (AUDIO: Something Inside) Later on, after being separated from his TARDIS for far too long, he lost them again while living on the planet Orbis. When Lucie Miller arrived, it took some time for him to remember. (AUDIO: Orbis)
War Doctor
After the Eleventh Doctor successfully saved Gallifrey from the Last Great Time War, together with twelve other incarnations, only the Eleventh Doctor retained any memory of this event, because "the time streams were out of sync". The War Doctor in particular believed he had destroyed Gallifrey using the Moment, (TV: The Day of the Doctor) causing the Ninth Doctor to punish himself for surviving, (PROSE: Meet the Doctor) refusing to take on travelling companions, by most accounts, (PROSE: The Eyeless) until Rose Tyler. (TV: Rose)
Tenth Doctor
When his TARDIS collided with that of the Fifth Doctor, the Tenth Doctor was able to resolve the crisis having remembered watching himself do so from the perspective of his past self. (TV: Time Crash)
Eleventh Doctor
The Eleventh Doctor felt he had lived long enough to forget his own age. Unlike the Tenth Doctor, who was still plagued by his actions in the War, he could no longer remember the number of children on Gallifrey at the end of the Time War, having "moved on". (TV: The Day of the Doctor)
All his memories were also transferred to his Ganger, made in his image. This meant both Doctors had the same relationship with Amy Pond, and the same basic consciousness, up to that point. The Ganger Doctor later sacrificed himself to stop another Ganger, Jennifer Lucas. (TV: The Rebel Flesh / The Almost People)
Before regenerating on Trenzalore, the Doctor remembered Amy Pond, "the first face this face ever saw." (TV: The Time of the Doctor)
Twelfth Doctor
After regenerating, the Twelfth Doctor could not remember how to pilot the TARDIS, to Clara Oswald's horror, (TV: The Time of the Doctor) nor any of his friends' names. (TV: Deep Breath) Later in life, all the Doctor's memories of Clara were erased, when both realised he had gone too far in her name, living through billions of years in his confession dial, dying again and again in a teleport-assisted loop, before staging a coup on Gallifrey, in order to pull Clara from her final moments. Afterwards, the Doctor told his story to a friendly face, who was in fact Clara, and did not recognise her. (TV: Heaven Sent / Hell Bent) This caused him to decide against erasing Bill Potts' memories, taking her on as a companion instead, during his time at St Luke's University. (TV: The Pilot) At the end of his life, the Twelfth Doctor was re-gifted his old memories, getting one last goodbye with Clara as a Testimony glass monument. (TV: Twice Upon a Time)
The Twelfth Doctor had some suppressed memories, which he could sometimes remember, like meeting the Cloister Wraiths beneath the Capitol as a child. He believed that memories became stories, once they'd been forgotten. (TV: Hell Bent) By his time, the Twelfth Doctor had long since lost count of the number of deaths he'd seen, or the number of people he'd killed. He believed that moving on was the best way to save more lives, since outrage would only cause delays, while threats still lingered. (TV: Thin Ice)
Thirteenth Doctor
As they regenerated into their next body, the Doctor relived memories from each of their past lives, recalling the magnificence of their TARDIS, (COMIC: The Many Lives of Doctor Who) Ian and Barbara, (COMIC: The Path of Skulls) the strangeness of some of their adventures, (COMIC: Card Conundrum) their love for London (COMIC: Invasion of the Scorpion Men) and for parts of Earth outside the UK, such as New York City. (COMIC: Time Lady of Means)
This regeneration, however, left the Thirteenth Doctor at a distance from her memories, while she tried to remember her own name again. Early on, she knew the word "doctor" was important, remembered her sonic screwdriver, and recalled having been "a white-haired Scotsman", but it took a face-to-face encounter with the Stenza Tzim-Sha to "knit [her] brain back together", restoring her memories. (TV: The Woman Who Fell to Earth)
After learning about her past lives as the Timeless Child, meaning her own history was not what she remembered, the Thirteenth Doctor suffered a major identity crisis. (TV: The Timeless Children) Resolving that she was still the woman she chose to be, "the Doctor", who opposed the Daleks, (TV: Revolution of the Daleks) she pursued her only lead, Karvanista, desperate to find out more about the Division. (TV: The Halloween Apocalypse)
When given the chance, the Thirteenth Doctor was willing to sacrifice her own life in a time storm to get answers. However, the Mouri forced her out. While here, she re-experienced one lost memory in her personal time stream, reliving the Siege of Atropos from the Fugitive Doctor's perspective. (TV: Once, Upon Time) After being taken to Tecteun aboard the Division's outpost, (TV: Survivors of the Flux) the Thirteenth Doctor was briefly able to enter the biodata module containing her stolen memories. It appeared to be an unstable structure. Swarm wrought destruction on these memories, however, in an act of revenge. (TV: The Vanquishers)
