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.