There is value in writing even if no-one reads the result. I would even argue that the main value of writing is in the process of writing. Writing helps you structure your thoughts. It provides you with a mental grip on the world.

This goes for non-fiction but also for fiction. As a fiction author I do relish being read and appreciated by the right people. However, the gratification I get from that is mainly an ego thing, whereas the gratification of producing something that I find complete and beautiful (that sense of rightness washing over you when you put in that final period) gives a deeper fulfillment.