What are the DACs that use good circuits in the Audio sections?

I just bought a Bryston BDA-2 DAC here on Audiogon, it has a discrete class A output circuit, as does the BDA-3 model.

Some of the Schiit DACS use discrete output circuits, like the Gungnir and Yggdrasil.

I'm sure there are others, but that's two companies that I know of. 

Discrete R2R DACs even in the absence of DAC chips still use logic chips. What's the problem with "chips"?

No.

There is no way to avoid a "chip" altogether.  Something has to do the processing.  That being said, Chord and PS Audio use a combination of hardware and software in the decoding process.  Chord chooses not to change OS through time, but PS Audio change OSs all the time which does have a distinct impact on the sound of the DAC.  These would likely be the closest to what you are talking about as they are not dependant on ESS/Cirrus Logic/Wolfson/Burr Brown chips you are talking about though other DACs don't use them as well.  

I personally had owned a PS Audio DSD with Bridge II for a long stretch and was quite happy with it.  Only sold it when I became a Chord dealer and now I use different Chord products at different times.  Both companies make superb products.