Infinite. The CRT inputs are analog. The Digital to Analog conversion is handled in your graphics card.
Right now the voltage is divided into 256 steps for each color and represented by 8 bits each, but that is the graphics card that does the digital part. I believe some graphics cards can do 10 bit which would give you 1024 steps for each color. For most of us, 8 bits is fine as we can't spot the difference between any of the 256 shades.
8 bit LCDs handle the conversion internally and the problem here is that sometimes the transitions done by twisting the Liquid Crystal are not as smooth and predictable as the CRT varying the voltage on the guns, so even though 8 bit might normally be adequate, it comes out banded on LCD.