doctor who - What would happen if you portalled the TARDIS inside itself?

The TARDIS is one of the few fictional objects that could contain themselves. Firstly, because "it's bigger on the inside" but secondly because the TARDIS has some means of sustaining paradoxes (which have their bounds, but it is capable to allow some paradoxes if you don't overdo it).

So if you had some sort of Portal technology (like in the game of the same name) which allowed you to create portals at will, you could shoot one portal inside the TARDIS and the other outside and fly through the outer Portal with the police-box-shaped TARDIS, assuming the portals were big enough.

If you remove the portals, where would the TARDIS be? Would this even work?
I realise canonic answers are probably hard to give, so I'm fine with some well backed-up hypotheses.

Answer

As @Daniel Roseman points out in his comments, this has actually happened in the Children of Need special mini-episodes Space and Time (contained on a Doctor Who DVD and confirmed to be "canon" by Moffet). In this episode, an emergency landing causes the TARDIS to materialize in the "nearest safe location", which happens to be inside the control room of the TARDIS.

Several interesting things happen once this takes place:

• 
  
  

  The doors to the TARDIS form a closed spatial loop. Anyone entering the doors of the "inner TARDIS" come through the doors of the "outer TARDIS" and vice versa. There does not appear to be any way to actually leave the TARDIS, nor for anything that's not already inside the TARDIS to get inside of it.

  
  


• 
  

  The "outer" TARDIS somehow drifts away from the inner one temporally: entering the "inner" TARDIS results in you coming through the "outer" doors several seconds earlier. Note that this does not happen immediately: at first The Doctor is able to move instantly between the doors, but seconds later Amy comes in from the future.

  
  


• 
  

  Eventually, The Doctor is able to reverse the effect through a "temporal implosion" in which both TARDISen dematerialize and everything goes "back to normal".

  
  

From this, given your portal scenario, we can draw a number of conclusion:

1. 
   

   It's perfectly legal for the TARDIS to be completely inside itself, and whatever temporospatial technology runs the TARDIS can readily adapt.

   
   


2. 
   

   Once the TARDIS becomes self-contained, it really is fully contained inside of itself. There is no path through space that starts or ends in the TARDIS except those paths that do both.

   
   


3. 
   

   There is still only one TARDIS, somehow recursively contained within itself. Note that Amy enters the "inner" TARDIS twice, while Rory only does so once, and yet both end up together in the "final" TARDIS.

   
   



4. 
   

   There is some form of temporal drifting effect between the inner and outer edges of the doors. Over time, the "external shell" (e.g. the outside of the doors) move ahead of the "internal shell", causing someone entering the contained TARDIS to come through the other set of doors in the past.

   
   


5. 
   

   Once self-contained, it is possible to remove the TARDIS from itself; this causes the TARDIS to dematerialize and reappear in "normal" space just like any normal take-off.

   
   


6. 
   

   Amy, a miniskirt, and a glass floor could potentially end the universe.