What is needed to control 200 LED's from an Arduino
With Charlieplexing, you can directly drive n*(n-1) LEDs from n pins. This means 210 LEDs from 15 pins. This can be done with a regular Arduino Uno if you use some of the analog pins as output. 

If you need different colors with different resistors, you have to do some smart segmenting. You also have to think about the time multiplex, high pulse current. This is probably easier if you know you will only have a few leds on at each time. 

See application here: http://www.evilmadscientist.com/article.php/bulbdial3