Has the Doctor ever observed himself, without revealing who he was?

Also known as "Has the Doctor ever observed himself, without revealing himself to himself?"

We see that the Doctor isn't afraid to work with himself to solve complex problems, but has there been any moments where the Doctor has observed his past/future self (time travel makes things confusing) without showing off he's the Doctor?

Answer

Yes, occasionally

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  He does this in series 1's "Father's Day", quite directly "observing himself" with Rose:

  
  

  

  
  

  He makes a big deal of not being seen by their past selves, out of fear of paradoxes; of course Rose goes and screws all that up, but that's neither here nor there.

  
  


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  This is broadly the point of series 8's "Heaven Sent"; the Doctor spends billions of years helping himself without really understanding anything about his situation (not until the last moments, anyway)

  
  
  


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  This is also, essentially, the plot of series 8's "Time Heist"; the Doctor et. al. are running through a script that had been previously set up by the Doctor himself, which he can't remember (and can't remember being responsible for) thanks to a liberal application of memory worm

  
  


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  Yet another Capaldi example, but series 9's two-parter "Under the Lake"/"Before the Flood" has the Doctor sending himself a message through time:

  
  

  
  

  Doctor: The messages my ghost gave, they weren't for you, they were for me. That list. Everyone after you was random, but you being the next name, that's what made me confront the Fisher King.

  
  

  ^{Doctor Who Series 9 Episode 4: "Before the Flood"}

  
  

  
  
  


• 
  

  While it's left deliberately vague, it's possible that he serruptitiously interacts with himself in series 8's "Listen"; Clara remarks that the note left on the chalkboard, alleged (by the Doctor) to have come from his "perfect hiding" companion, is in the Doctor's own handwriting:

  
  

  

  
  

  
  

  Clara: It looks like your handwriting.

  
  

  Doctor: Well, I couldn't have written it and forgotten, could I?

  
  

  ^{Doctor Who Series 8 Episode 4: "Listen"}

  
  

  
  

  This one is admittedly debateable especially since, as Clara notes with her next line, it's entirely in-character for the Doctor to have left himself a note which he then forgot about.

  
  
  


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  We might also consider the Valeyard, from season 23's "Trial of a Time Lord" arc. The Valeyard is revealed (by the Master, not by himself) to be an amalgamation of the Doctor's darker impulses, something of an in-betweeny, timey-wimey, pseudo-future-regeneration...thing:

  
  

  
  

  Doctor: Just a minute! Did you call him the Doctor?

  
  

  Master: There is some evil in all of us, Doctor, even you. The Valeyard is an amalgamation of the darker sides of your nature, somewhere between your twelfth and final incarnation.

  
  

  ^{Doctor Who Season 23 Episode 4: "Trial of a Time Lord: The Ultimate Foe"}

  
  

  
  

  The Valeyard has previously spent the entire season-long arc acting as prosecutor at the Doctor's trial, without feeling the need to reveal his identity.