Technically, as Rory points out, you can’t go outside the universe. The universe is all that there is so by definition there isn’t anything else to go ‘out’ into. That being said that isn’t any reason why you need that to stand in the way of a good adventure.
In ‘The Doctor’s Wife’ the Doctor is only able to achieve this feat by burning up TARDIS rooms to give them extra power. Previously he has ejected parts of the TARDIS, presumably to shed weight, but this is the first time he has converted them into extra energy.
Is this because of the distance travelled, if so why not travel back in time when the universe was smaller? Is there some barrier that must be passed, or a field (gravitational or otherwise) that makes it difficult to go outside the universe?
Whatever the reason leaving the universe is difficult, not something that should be undertaken casually. Such trips should be rare and are not taken without great cost. The Doctor does so here because he is hoping to find that one of his people.
Visually the outside of the universe appears to be the same, a black void rather than white emptiness. It might be that during ‘The Doctor’s Wife’ we are just looking back towards our universe and that if we looked the other way we’d see something strange and incredible.
If we take it that most living beings evolve in response to their surroundings it stands to reason that they’d be unable to comprehend or even register stimuli produced from outside our universe.
This takes us into Lovecraftian realms, where the exterior of our universe is inhabited by invisible entities, unknown colours and sounds that can drive people mad. This is a breeding ground for the impossible.
The planetoid that the TARDIS lands on is described as a junkyard. Large sections consist of the dead TARDISes lured there by House but what about the rest? Is this where space debris eventually ends up, after eons of travel?
Distance might not even be a factor. This could be where things that fall through the cracks in reality (such as encounter by the Doctor throughout season 5) end up.
Player characters might discover remnants of ancient alien cultures, whether it be ships or fragments of their world. What wonders were created by species that pre-date the Time Lords?
Since there is something that our universe is expanding into it might be a previous universe. This might be the natural cycle, with each new universe emerging and consuming the one before it.
This would mean that relics found needn’t have originated in ours. This has huge implications, revealing the existence of species before the Big Bang. This could be the birth place of Old Ones, Eternals or Guardians.
In this distant realm there still might survive ancient beings, slumbering or waiting for their chance to emerge. The presence of the player characters might provoke them into taking action.
There could even be those who wish to awaken this beings. It isn’t hard to imagine nihilistic cults making the difficult journey to outside the universe to bring about doomsday. This would be reason enough for the player characters to pursue them.
The outside the universe might also be outside the influence of the time vortex. It could have been a way for species to escape the ravages of the Time War. While their might not be any Time Lords other refugees could be found, oblivious to the end of the conflict.
If this area is truly timeless then player characters might discover species that no longer exist due to time being rewritten. What would the result be if they returned home? Would this create paradoxes? Would they encounter alternative version of themselves?
We might discover that parallel dimensions aren’t layered but separate universes existing side by side. In the past the TARDIS has simply ‘hopped’ between the two, inter-universal wormholes bridging gaps between N and E space.
This gives more weight to parallel realities. They are not on their own but neighbours, giving player characters a much bigger playground to explore. Everything that happens in any reality matters because it is all equally real.
If all these universes are also expanding then they would eventually begin devouring each other. It could be this process that provides the necessary energy to fight off entropy. It could also mean that eventually there will only be one super-universe that has absorbed all the rest.
If the inhabitants of these universes became aware of this this could lead to conflict, as they try to ensure it is their home that survives. This could be the beginning of an mind bending campaign with parallel realities fighting. Player characters could take sides or find allies (including good versions of species such as the Daleks, Cybermen and so on) to find a better solution.
Another way to use the idea of neighbouring universes is to focus on exploration. If the Time Lords knew there was an ‘outside’ to our native universe wouldn’t they have sent scouts to map the area and make sure there weren’t any threats?
The player characters could be assigned a specially designed TARDIS, suited to the rigours of inter-universe travel and sent out to boldly go where no Time Lord has ever gone before.