In mid eighties, when I used to visit NY Manhattan, there was a fashion to ride fixies with no brakes. Pretty well all town messengers, delivery boys rode them. In those days, it was total anarchy on bikes in Manhattan. You rode without any regard for red light or sidewalks, just watched your a$$. Red just meant you needed to be extra watchful on the lookout.

I got one too and on my next few visits, joined in the anarchy. I was then in my early twenties. That was great, until some years later, city cracked down on that lawlessness.

Mind you, I kept the front brake and would have been considered a sisi, but I didn't really need it and if you are good, you don't need it. Just need to ride like you would play a chess game - plan and look ahead all the time and be in total awareness, no daydreaming when you are moving, in the city traffic that is. Riding fixie outside a town is probably not much fun and I'd say it limits you unduly, never done that myself.
You get told, that's stupid having no brakes and yes, it is. But to get proper adrenaline flowing, go without. It can be compared to free rock climbing without a rope. Only don't get cocky, it pays to keep humble while doing those crazy things or you will get in trouble.

For those not familiar with fixies, you never need to put your foot down on ground, even standing still (like they do it on a velodrome). When you get good and you come to a stop, you just stop and stay still sitting on the bike without balancing by moving front wheel (as you need to when you are new to it), just sit and even can look around.

Aah memories, get me twitching to get fixie again