We know that the Borg's aim is to assimilate all worthy species to attain perfection. But the thought struck me: what then? The Borg seem to preoccupy themselves with assimilation, and this takes up a fair amount of their time, as we see in Scorpion Parts I and II when they try to assimilate Species 8472. But, even if they do succeed at assimilating everything they want to and achieving a state of 'perfection', what is their goal then?
The Borg have 2 goals, in my opinion.
To obtain perfection, either biologically, or technologically. This goal is something that may or may not even be accomplished, as the further you improve, the more you may understand, and realize you can improve even further.
To create order out of chaos. This seems to be the underlying goal, to their quest for perfection.
Seven of Nine: 300,000 individuals have been transformed into drones. Should they be congratulated as well?
Borg Queen: They should be. They've left behind their trivial, selfish lives, and they've been reborn with a greater purpose. We've delivered them from chaos into order.
Source: IMDB (Internet Movie Database)
In 2378, a crippling blow was delivered to the Borg when Voyager discovered one of their transwarp hubs and destroyed it, killing the Borg Queen (again) and devastating the Unicomplex in the process. During this battle, the Borg were infected with a neurolytic pathogen, which was carried by Admiral Janeway and designed to disrupt the hive mind, to bring chaos to order. It was this pathogen that killed the Borg Queen, and allowed Voyager to destroy the transwarp hub.
To create order out of chaos is a massive goal, and is not contained to simply our galaxy, but the entire universe. Given the chance I believe this is the Borgs end game, to attempt to bring order to the whole universe, something that may very well be impossible, but would be a worthy goal to work toward indefinitely.
For what can bring order out of chaos? Why, only a perfect entity of course, The Borg.
They'd probably just wait around...
It's surprisingly tricky to find anything online pertaining to the Borg's long term goals post galactic assimilation, but its reasonable to assume that as the Borg ignore lesser civilisations which offer no technological advancements, eventually they'll begin to ignore more and more of the galaxy until they have no other option than to just wait around for someone worth assimilating to come along. They'd probably become the most accomplished explorers in the galaxy, they've already travelled to fluidic space.
I could even imagine they may begin to encourage different species to evolve and develop in order to assimilate them when they had something to offer.
Although the issue seems complex, it actually is not. Assimilating everything encountered would provide enough information to formulate a new primary objective...that could be anything, including time-line related or multi-verse related (from a ST perspective) or even multi-D.
It's just like asking a human what is his purpose in life. Very very few will provide a worthy answer...and of course, after that you can ask them what's next.
Realistically speaking, even if one's primary objective is completed, there are actually many more things to do or improve upon and one may find a new target objective as soon as the primary is complete.
Perfection for the Borg would mean having the perfect collective with the ultimate technology. But as I said above, that could grant them access to other things unknown until the point of maximum technological assimilation.
There's more about species 8472 assimilation in the story line episodes in Star Trek Online. You should try them, it takes only a few hours to complete them all.
As far as I know there is no canon answer so the following is surmise
The Borg pursuit of perfection is an infinite loop. At no point can the Borg actually say we are now perfect. If the Borg did decide they were perfect, there wouldn't really be a goal except to be perfect, whatever perfection means to them.
However, if at some point the Borg did assimilate everything in the universe that they deem worthy of assimilation, the likeliest scenario is that they would simply go about their business until a species that they deemed unworthy of assimilation in the past achieved some technological or biological distinctiveness, then they would assimilate it.
The scope of assimilating everything though boggles the mind especially if you consider the mirror universe and possibly other alternate dimensions, not to mention that alternate universe may have alternate Borg empires.
xelor's answer wraps it up nicely in five words: "The route is the goal."
The question presupposes a goal-oriented imperative. The pursuit of perfection itself could be its own goal the same way counting rosaries is for a catholic or meditation is for a Buddhist. Revere perfection by pursuing it.
Assimilation might also be an instinctual imperative like eating or procreation. Ancient human history offers examples of hedonist and/or pagan societies who elevate these imperatives to religious importance. Looked at this way, the borg assimilate because it's in their code and/or DNA and rationalize it with a religious assertion (The Assimilation Assertion).
In these contexts, the question is unanswerable. (Why did the chicken cross the road? So he could cross back again.)
I'm inclined to believe that if the Borg were to continue expanding they'd reach a critical point where they would condense into separate (identical?) entities - effectively undergoing mitosis.
Another possibility may be their inevitable expansion into isolated realms of subspace or whatever necessitating emergence of new Queens.
But before then, during their expansion, I believe they might just evolve into a 'generic all-consuming homogeneous mass', graduate onto eating stars, then go into 'hibernation mode'.