Sooner or later all Time Lords must face the fact that they are a limited number of regenerations. Eventually they’ll be in their 13th incarnation, the end of the road. The Doctor is lucky enough to get a whole new cycle of regenerations in ‘The Time Of The Doctor’.
If your group enjoy taking the role of the Doctor himself that gives you plenty more incarnations for them to take. It is very unlikely that any player, within their lifetime, will actually see what the 2oth or 25th Doctor will actually be like, giving you room to create their adventures without fear of being contradicted anytime soon.
We’ve seen as well that the Doctor can be affected by the actions of his future incarnations, whether it be meeting someone who has met his next incarnation or finding the site of his final battle.
Now there are many more Doctors, influencing events. This can be a good way to foreshadow those distant incarnations, giving small hints about their personality and how they operate.
If you aren’t intending for these later incarnations to be playable these future versions can be very fanciful. Remember how the 10th Doctor found River Song’s description of his 11th incarnation as impossible. Now consider how different he might be in his 16th incarnation.
What about a Doctor who has merged with his TARDIS, capable of changing his external appearance and travelling through the vortex with just a thought? He appears in times of need and vanishes the moment the problem is resolved.
Consider a Doctor who realises that protecting a single universe is too small scale and protects multiple realities. No longer will divergent dimensions be threatened by the actions of someone in another dimension, as happened in ‘Stolen Earth/Journey’s End’. Would such a Doctor forget where he came from?
How about a Doctor who no longer deals with the Daleks, Cybermen and other monsters but wrestles with gods themselves? This Doctor challenges Eternals, primal evil and other powerful entities. A more powerful version of his 7th incarnation who plays chess not just with peoples lives but all of time. Could such a Doctor be in danger of becoming the very god-like being he opposes.
The possibilities are virtually endless and it isn’t just the Doctor who might come across his future handiwork. PCs might learn of the actions of a Doctor the players have never heard of. They might even meet a future incarnation. What will they make of this strange and distant being?
At some point the Doctor will become the Curator (probably his last incarnation). In ‘The Day Of The Doctor’ the Curator hints that the Doctor will revisit some old faces, so there might be future versions of the Doctor that wear the same face as previous Doctors.
This could cause some confusion, even for the Doctor himself. A time traveller might meet the 3rd Doctor only to discover it is the 19th. If their personality is different then this could only add to the confusion (not to mention trick players into thinking you’ve made mistakes in continuity). A future Doctor might take advantage of this by enter the time stream of their previous incarnation to have a few more adventures with old friends, without them realising he isn’t the Doctor they know.
Within your own campaign a Time Lord PC reaching his last incarnation may want something similar to happen, so that they can keep going. This really depends on how exactly a Time Lord gets a new regeneration cycle.
In ‘The Time Of The Doctor’ the Time Lords send a cloud of golden light (usually associated with regeneration) which enters the Doctor’s mouth. As discussed in the previous article this could be the raw energy he needs to trigger a regeneration but there must be more to it than that, to overcome the limit.
It could be that the limit is physical, that a Time Lord’s body is only physically capable of withstanding the trauma of completely changing 12 times. After that the process that triggers a regeneration burns out.
The energy given to the Doctor could be special, in that it not only heals his physical attributes but the part of his body that is responsible for regenerating. Now reset he is capable of 12 more regenerations.
The limit might be mental. In that a Time Lord is physically capable of transforming much more than the 12 limit but their mind doesn’t activate the process. This could be similar to how the Master wills himself not to regenerate in ‘Last of the Time Lords’.
If this is the case then the energy could be coded, to give the Time Lord a signal that they are allowed to regenerate. This could explain the look of realisation that the 11th Doctor gives after receiving the energy.
After receiving the new cycle the Doctor regenerates but he was already dying of old age. It might be possible to receive a new cycle without triggering a regeneration or it could be that the energy forces the body to change immediately.
The new cycle is provided by the Time Lords but is it possible to obtain it through other means. One would think that such information would be kept secret, to prevent any Time Lord giving himself unlimited regenerations.
Still, it could be recorded somewhere to prevent that information from being lost forever. PCs could go on a quest to find it and decode the information. The data they find could in turn send them on further quests to complete the stages needed for new lives.
A desperate Time Lord could experiment to try and discover the process for themselves. This could be done by a Time Lord PC or the PCs could have to stop the evil experiments of a NPC who is going too far in his desire for immortality.
What we don’t know is how many new cycles can a Time Lord be given. When the Doctor reaches his last incarnation will he be given another 12 lives and so on? How many versions of the same man can there be before even a near infinite expanse of space and time is filled?
Bigger On The Inside 
It seems worth pointing out that the issue of extra regenerations has been brought up in classic Who for the Five Doctors special. The Master, by this point short on regenerations, is promised an additional cycle for his service (suggesting that they are doled out in batches of 12 and that the High Council is capable of offering them), but the secret of unlimited regenerations is the goal of the quest for Rassilon’s tomb (suggesting that there’s a natural limit eventually). The secret of infinite regenerations is supposedly known only to Rassilon, but it turns out to be a trap for the power-hungry, so it’s unclear if it actually exists.
I would suggest that a sufficiently great service (read: character arc + major quest conclusion) would be an appropriate point to offer an additional cycle to PCs, and The Five Doctors offers sufficient precedent to do so even absent circumstances such as The Time of the Doctor. (If it can be done in a reward ceremony rather than as a deus ex machina, you could even have someone refer to the precedent in an official pronouncement.)