What is the most analog sounding CD player under $

i was very impressed with the vincent s1.1, which is a hybrid tube player--very natural sounding. they also make tubed players (probably available used at your 1k target) which are well worth checking out.

Meridian 508.24 is one to consider

Perhaps a Rega Apollo?

I have a Rega Saturn which is absolutely wonderful - detailed, neutral, and smooth. You might be able to find one for around $1k used. New price is about $2,500.

The Apollo costs $1k brand new. By reputation it is a little warmer and more analog sounding than the Saturn, though possibly less detailed than its more expensive sibling.

Rega's digital equipment has a reputation for sounding more analog than most.

The Rega is wonderful. I'm not sure I'd call it "analog" sounding. Very realistic and detailed. If you want to warm it up, there are good tubed models out there such as the Raysonic 128 or the Shanling 3000.