Do I understand correctly that your concern is about the logs? You can set up an Amazon S3 static website, and no logs will be collected, unless you enable it. The first year is free, and after that it's pennies a month. I use this, plus Route 53 DNS, and my total bill each month is ~ $0.54 (US), with $0.51 for Route 53.

If you need other options to cut cost, freedns.afraid.org offers free subdomains under ~68000 second level domains.