I’ve been using fastmail.fm for over five years and have never seen an ad.
It is not free, but the cost is minimal for having just great e-mail.
I used fastmail.fm ages ago and the service was excellent - also free, but it sounds like they changed their business model.
Anyway, there is no free lunch. It’s either ads of some sort plus potentially have the emails parsed for keywords and also archived, or pay a few bucks.