‘The Day Of The Doctor’ allows us to examine what happens when the same Time Lord meets other incarnations of himself. This is dealt with a little more indepth that ‘The Five Doctors’ which never really addressed the issue.
Firstly the 11th Doctor is the only one who begins to have a vague recollection of the wormhole when it appears and realise that he is responsible for throwing the fez through (and then himself).
However he believes that this is interrupting his current adventure in the under gallery and subsequently doesn’t remember what happens next, blaming this on the 10th Doctor not paying attention.
We might see this as evidence that what the Moment does is altering the timeline. The 11th Doctor only remembers the wormhole because the 10th Doctor has just seen it ‘now’ and those memories are only just catching up to the 11th. The problem is why doesn’t the 10th Doctor remember the wormhole and a fez coming out, as happened with the War Doctor?
If this is part of the original history we can explain this away as the 10th Doctor’s amnesia resulting from the time stream being in flux. His last memory is the wormhole and the fez and once the 11th Doctor emerges he can’t remember anything further (to the conclusion).
The 11th Doctor will always have that memory of the wormhole (and possibly that someone emerged. His love of the fez may have its origin in that fragment of memory, with a nagging suspicion that he will be involved in the resolution of the matter.
When the Doctors are placed in a cell it is apparently their close proximity that is worrying. This makes sense as multiple incarnations of the Doctor could exist in the universe at any point, without causing problems.
Since they weren’t concerned about their proximity in the woods it might indicate that the enclosed space is the problem, rather than the distance between them. Alternatively it could be prolonged proximity that is the problem.
It is not made clear what kind of anomalies might occur. Notably the anomaly is considered ‘nasty’ rather than ‘dangerous’ or ‘lethal’. This might indicate that it will just result in further problems that could further destabilise history rather than kill the Doctor or end the universe. That being said the Doctor is sometimes a master of understatement.
In your own games the existence of anomalies might be evidence that the same Time Lord or other time traveller is too close to his other incarnations. A PC Time Lord could realise that this indicates that his own past or future version is nearby.
‘Invasion of the Dinosaurs’ and ‘Fathers Day’ provide some good examples of time anomalies. There are aspects of these anomalies that could be advantageous to a Time Lord. It is easy to imagine the sudden appearance of a time shifted dinosaur would provide the Doctors the distraction they need to escape.
We are presented with another distortion of time when the War Doctor has his sonic screwdriver begin a calculation, with the 10th Doctor confirms is running in his and the 11th Doctor finds has been completed in his.
Once again we can view this as an alteration of time or something that was always true. If it is the former then we might review how the presence of program may have affected its capabilities in the 9th, 10th and 11th Doctor’s subsequent adventures. If it is the latter then we can assume that they never noticed (but in adventures set prior to ‘The Day of the Doctor’ it now becomes vital the screwdriver not be destroyed).
Most importantly the consequences of the War Doctors actions are instantly reflected in the 10th and 11th Doctor’s screwdrivers. At that point, in the cell, time was taking it for granted that both the War Doctor and the 10th Doctor would eventually become the 11th Doctor.
This can be an important point, especially if the PCs are also expecting to see an outcome based on a presumed turn of events. If the 10th Doctor or 11th Doctor’s screwdriver hadn’t shown the calculation it might mean that the changes weren’t being reflected, that something was going to destroy the screwdriver or at least stop it running (maybe the 9th Doctor who would have no memory of these events) or worse still that the War Doctor or 10th Doctor weren’t getting out alive.
This mystery can keep players guessing and prevent anyone taking things for granted. Of course you then have to explain why it didn’t work at some point. Nonetheless it shows how failing at an action or plan can still create entertainment.
When the three Doctors depart it is said that they won’t remember what happened because their time streams are out of synch. This handily explains why previous multi-Doctor stories (with the exception of ‘Time Crash’) didn’t have the oldest incarnations of the Doctor having memories of these adventures, several times over.
Is this really a result of the time or could it be a mechanism of the Time Lord brain? It could be that the Time Lord knew the danger of meeting your future self and the possible paradox of learning information from them. Thus any such knowledge is locked away.
Whether through nature or design it is possible that some memories still leak through, just as the 3rd Doctor apparently knew that 4th Doctor would look like Tom Baker (or he was just very good at interpreting Sarah Jane’s gestures).
This can be an excuse a player needs to explain why their Doctor recognises something that a later Doctor experiences. It stops becoming a continuity error and is instead an indicator that this was mentioned when the Doctor met a future incarnation.