The Doctor has been leading a nomadic existence ever since he stole an obsolete TARDIS and left his home planet of Gallifrey far behind him; it’s only natural, since his TARDIS gives him the ability to travel anywhere in time and space. After all, why would anyone restrict themselves to one place if they had a whole vast universe to explore and seemingly unlimited time in which to do it?
All of the greatest spectacles that ever were or will be are easily within his reach. He can visit any point in history, on any planet he chooses, and even live that history if he so desires. He can watch kingdoms rise and fall, civilisations grow, develop, and find their way out into space, expanding, colonising, and sometimes conquering.
He can bear witness as stars and whole galaxies are born, observe the formation of planets, marvel as life upon them evolves and gains sentience, and gaze in wonder as mountains rise, only to crumble into the sea millions of years later. In his centuries of travel, he’s seen lush valleys transform into bleak and barren deserts, and seas evaporate or turn to ice. Nature is beautiful and strange, but can often be harsh and cruel.
He has long since lost count of all the famous historical figures he’s met and influenced, for good or bad; rulers, statesmen, artists, composers and writers, great warriors and noble heroes, poets and dreamers, lovers and saints. Some he’s liked, some he hasn’t, but all have been interesting and he doesn’t regret his encounters with any of them.
And so he travels, from city to city and world to world, one time period to another, revelling in the sheer variety and complexity of sentient life. He helps where he can, educating, and guiding, and once in a while, destroying, because not everything that comes to be is something good and wonderful and deserving of life. The universe is not that kind or benevolent; if it were, then he would not feel needed and his life would have far less purpose. As it is, trouble and adventure are never hard to find, and through it all, there are always new things to see, places to discover, and interesting people to talk to.
It’s an exciting way to live, filled with such diversity that boredom is rare. The universe is his oyster and he’s as free as it’s possible to be, perhaps freer than anyone else has ever been, with nothing to tie him down. He can go everywhere and anywhere; no place or time is beyond his reach, with a single exception.
The only place he can never return to is home, because Gallifrey is gone.