For Variety, Fun and Challenge...
Built up my first fixed gear conversion three falls ago for a winter riding treat. It turns out that I loved the change, after having ridden geared bikes for many years.
Fast forward to today, and I still enjoy the simplicity and experience of riding fixed. It's fun, plainly said. I also enjoy the challenge of mastering new skills, so riding fixed has given me opportunities in that way.
Finally, having a fixed gear bike among my others offers variety any time I want to change the ride that day. Have a standard rolling hilly ride that you enjoy? Do it one day on a racing bike, the next on a touring or utility bike, and the third day on a fixed gear. Three completely different experiences, but on the same basic course.
Want to tour the city? Why not on a fixed gear?
Errands to run? Using a backpack offer enough storage? Use the fixed gear instead of one of the others.
That sort of thing...