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

Audio Note 

+1 @roberjerman re: Lampizator

As mentioned above PSAudio, Chord and some others use software for the digital conversion but I have no idea if there is a chip somewhere in the signal path. They might use a chip on the digital inputs. CD player or streamer will have chips, not sure you can do digital and avoid using a chip somewhere in the audio path. If you stream from the internet there are chips in the modem that covert the analog to digital. 

To make it a bit more clear, I am not talking of Chips/etc. in the units other operating parts, but in the audio signal path...what you hear, not what makes the display work. 

ARC preamp have chips in them to run switches, drive display, count tube hours... as do many many others including Aesthetix, the display is defeatable via the remotes for clearly better sonics. The Aesthetix Pandora DAC has an IMO stellar analog output stage ( as does the Lampizator... I own both )....

I don't know of any DAC that doesn't have a chip or PCB in it. 

chips, a concern?  Well, that is up to you.  I assume that if chips were SOTA, then Audio Research, Conrad Johnson and more would have pre and power amps that were all really big chips. 

 Chips/I.C. etc.  are in the ear of the beholder. 

Confused about your question but yes many have good analogue stages. For Example Modwright Elyse is built on a PCM1794 and has a tube analogue stage after conversion.  
Here is photo:
http://www.modwright.com/cms/resources/elyseint-w1000h680.jpg

Lampizator with a tube output stage is a good choice! Expensive, though!

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.  

No.

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

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.