Some write for the beauty of it. When I read a beautiful paragraph, I get a warm feeling of appreciation. Admiration for the intellect of someone capable of bringing out something so beautiful or smart. That makes me also want to try it, the same way as seeing someone complete a task in a masterful way, makes me at least want to try it myself as well. If for nothing else, at least to test how easy or hard it is. This is called writing in chase of beauty.
You will find that most things are not easy to do at a high level, yet writing does not seem hard for the non-writer. The levels of it are hard to distinguish if you do not dive deep into them. If you do start writing, you quickly find that making sense is hard. Making it good is even harder. This is called writing for the challenge.
I write because of the intellectual stimulus as well. It scratches that itch. Philosophically, it also scratches the itch of playing with the probability of creating something beautiful, that will make my name live in eternity. I will be a rediscovered writer that was not appreciated in my lifetime, and yet my followers will devour my works in 100 years from now, and wonder what kind of life I led, what life could have created this primordial soup out of which my books came out. What a sweet fantasy. Writing for the intellectual admiration fantasy.
Why do you write?