In answering this question you have to consider that the Borg are not individual entities, all Borg are connected into a collective and have distinct roles and tasks they are assigned to do. You also have to consider what there ultimate goal is. As viewers the only thing we absolutely know is that they have a single minded goal to achieve "absolute perfection" If you look online there are a number of theories about this, none of which have been confirmed as canon but the most popular one is that the Borg will not be happy until they have assimilated everything that they see as worthy of assimilation in the galaxy (and in scorpian we see that desire take them to fluidic space). Or destroyed anything that is seen as inferior. The reason for this is that the Borg are incapable of true innovative thought, they rely on the knowledge that they assimilate from other species and then apply that knowledge to their own technologies rather then engaging in indepth research on blue sky ideas. Even the Harmonic Resonance chamber was created after the correct technologies where assimilated in and then applied.
As the Borg all serve the collective, and as the collective believes in complete efficiency then you can assume that all borg, no matter what they are doing, are serving this greater goal. Yes they have roles outsode of just assimilation. as an example
There are Borg assigned to protect and patrol the edges of Borg Space hunting down anyone who may attempt to invade, attack or simply wander in.
There are Borg who will be searching for and collecting any cubes or drones that have been identified as having been damaged, destroyed, captured or lost.
There are Borg who are working soley on building the Borg infrastructure, manufacturing the borg ships, transwarp conduits, combing through the knowledge and technology assimilated to identify what is worthy of adding to the collective and what knowledge is not. Attempting to understand technology that they cannot investigate.
There are Borg who's role is to act as scouts, searching for and identifying the next targets for assimilation.
And there are Borg who are being sent on missions to assimilate and learn, such as cubes sent to the Federation.
But there is no concept of Borg free time, if a Drone is not required it sleeps going into hibernation until needed. If a Cube is not required it goes dark drifting in space until required.
So what do the Borg do when not assimilating, lots, but all is aimed towards that singular goal. The assimilation or destruction of the species of the universe the collective and no single borg would ever do anything that did not directly or indirectly help serve this goal.