Something like a dsPIC33 has 21 programmable digital output pins. That'd work. You'd need a D/A for each line out.
I'm just getting into this area, but could you just run the digital signal through a PWM to get the anolog outputs needed? Elemenating the need for costly DAC's.
You can do this. The limitation is clock speed. For example, if you want 16 bit resolution PWM at 41.4kHz sample rate, then you need to be able to modulate the pulse width in 1/[41.4k*(2^16)] second increments. This comes out to 368 nanoseconds. This means you need a 2.7GHz clock, minimum. Since a processor probably can't do this with one instruction, and that instruction cycles take multiple clock cycles, this implies the need for a 10GHz clock to do this on one pin.
So, before I look dumb, this can be done with reasonable clock speeds using multiple pins. Let's use two pins. Viewing these as a pair of 8-bit registers, we can represent 16-bit audio with two pins. This requires an 81ns clock, which is quite reasonable. We could probably get an acceptable sample rate with a dsPIC clocked at 40MHz. Because of the way the PWM registers work, I think you can get the pulse turned on and off within a clock cycle, but almost certainly an instruction cycle.
If a clock cycle, you can get a 25 ns pulse width, otherwise it will be 100ns. In both cases a 41.4k sample rate is possible. The "high" byte represents numbers up to 2^16, while the low byte represents numbers up to 2^8. One needs to apply a 1/256 resistor divider to the low byte pin and add to the high byte pin in hardware.
Now to make an A/D converter out of this, you feed the output of the D/A into one side of a comparator, and the audio signal into the other side. You then set up a control scheme to cause the output of the D/A converter to track the audio input; for example:
Of course, this implemented exactly would require too much CPU to keep up, so one would have to apply some algorithms for quickly resolving the value. This is as though we're implementing a unity gain op amp with digital innards, where we harvest the digital representation every 1/(sampling_frequency).
I think some PIC chips have comparator registers built-in. This would reduce external circuitry to resistors and capacitors. You may want to apply an active filter for anti-aliasing. A passive filter may end up needing a cut-off frequency below what's acceptable for getting adequate rejection at fs/2.