None that operate legally will be able to avoid logging chats when ordered to do so.
For example OpenAI were required by a US federal judge to log all chats, and make them discoverable to lawyers representing The New York Times last year. https://www.businessinsider.com/openai-new-york-times-copyri...
Additionally the company can be gagged by a court from disclosing that the chats are being logged, at least in the USA and the UK.
The Lavabit case years ago was quite scandalous, things have only gotten worse. There should have been much harsher limits on what companies can be compelled to do.
Although a good lawyer can appeal a board order. What the courts will say is unknown, but there are real constitutional questions about ordering everything.
There is a legal distinction between document retention, which is what OpenAI was ordered to do, versus re architecting to generate documents for logless providers.