When Adam has the same type of computer chip installed the Editor has instant access to all his knowledge about the Doctor including that he has TARDIS. He implies that he will do the same to the Doctor to gain all of his knowledge, which is supposedly infinite.
Presumably all of this is based on the information that Adam has on the Doctor, which must be very little. Therefore it might not be literarily true that Doctor has infinite knowledge, that is just Adam’s impression of him. We must also wonder just how much the Doctor told him about his travels for the Editor to consider the Human Empire tiny to what he has seen.
We later learn that the Daleks were behind the installation of the Jagrafess so it could be that all they needed to learn was a little background information to recognise him as their masters old enemy.
‘About Time, Volume 7’ by Tat Wood theorises that the Doctor might be able to access a database of information via the TARDIS, either stored in its memory or gathered from its telepathic circuits from the local population. Evidence of this can be seen in the way that the Doctor ‘searches’ his memory to identify the Slitheen and his later recall that Harriet Jones is a prominent historical figure.
The 8th Doctor novel introduced the idea that in order to survive the destruction of Gallifrey the Time Lords downloaded their memories and the Matrix into the Doctor’s mind (deleting some memories to make room).
Either scenario could explain why the Editor believes the Doctor has access to a vast amount of information. If they implant a chip to access his mind, they too will have this expansive knowledge.
There are many incidences of the Doctor not knowing things so how do we reconcile this with his supposed omniscience? A clue can be found in ‘The Beast Below’, where the Doctor is unable to explain to Amy why he is checking a glass of water and he confesses he doesn’t know because he thinks so much that its hard to keep up.
In ‘Silence In The Library’ the 10th Doctor bemoans the fact that his head is to full of stuff and wishes it was bigger. He is clearly annoyed that he knew something but wasn’t ale to get at it because of all the other stuff he knows.
Even if the Doctor does have near-infinite knowledge it doesn’t mean he can access it readily. He needs to shift through it, narrow his search criteria and dismissing data that isn’t pertinent to the current situation.
When the Doctor is relaxed he is able to interact more normally, enjoying the current situation. This can lead to him not knowing things, such as whether ‘Aliens of London’ is when humanity makes first contact with alien life.
The Doctor tends to become more distracted when he is searching his memory. He tells himself to think, mumbling to himself as he runs through the data flooding through his mind.
This behaviour clearly shows two states. One where his access to knowledge is ‘off’ and one where it is ‘on’. His companions are often necessary to switch him between these two states, their questions further helping him process what he is learning.
In game terms this is a great example of how to handle Time Lord characters. They have ready access to their skills but by spending Story Points they can ‘know’ important information to the plot.
If the Doctor really is gathering this information via the TARDIS then it is possible that he could share this gift with his companions, just as he does the gift of translation. A Time Lord could take this as a special good trait.
This would allow any companion not only able to speak any language they encounter but know useful things about their surroundings. A companion who has ready access to cultural and historical data is much more likely to keep out of trouble.
Adam’s abuse of his own future knowledge could be why the Doctor doesn’t do this. Even the most trusted companion might use the sudden influx of knowledge to their advantage or learn something they shouldn’t.
It could be that human minds just aren’t capable of handling the knowledge. It obviously affects the Doctor’s behaviour, who knows what might happen to a simple human. Like Donna they could be at risk of burning their mind away.
Companions would also be much more likely to realise that the Doctor is interfering with their minds. It is one thing to be able to understand any language, it is quite another to have strange knowledge implanted in your brain.
The Doctor is used to his companions never questioning the gift of translation. The 4th Doctor realised there was something amiss with Sarah Jane when she asked about it and the 9th Doctor was surprised that Rose was upset by it.
It would be much more apparent to a companion that their mind had been meddled with when they suddenly knew things they couldn’t possibly have known themselves. This could be a very frightening experience, especially if the knowledge affected their behaviour.
If you instead go with the theory that the Doctor (and maybe your Time Lord PC) has part of the Matrix downloaded into their brain then they too might question how it affects their personality.
The Time Lord PC might hear the voices of the Time Lords in their mind. When they regenerate this could even be an opportunity for one of those trapped personalities to assume control.
Alternatively the Time Lord might make those memories ‘sleep’. The stored Time Lords might slumber or exist within a psychic landscape, such as the one in ‘Timewyrm: Revelation’ or the garden in ‘The City of The Dead’. A representation of the Capital might make a good place to house them.
While the Editor might make threats it is doubtful that he possessed technology capable of streaming ‘infinite knowledge’. The Doctor has trouble accessing even a small part of it, downloading everything would likely be very destructive.
A Time Lord could take this as a special good trait to protect them from psychic assaults. if another entity or machine was able to breach their mental defences raw exposure to the sheer amount of knowledge they possess could be very painful, if not lethal.
Having access to all of this information can be blessing and a curse. It can be an ongoing character arc for the Time Lord, coping with his burden. Might he consider removing it and how will it affect him if it was taken away?
If the knowledge provided is second-hand what happens when he finds that the information is wrong? Is it an indication that time is wrong or that he can’t depend on the data he has?