The PHP requirement is going to be tough; and tough for someone to provide for free sans ads.
Google Sites can use Google App Engine for dynamic sites but that is Python.
Amazon has (I think!) a "free" tier for their EC2 virtual computing, so you might be able to set up an entire minimal LAMP OS (they have CentOS and Debian and Ubuntu and the usual suspects as options) and serve everything using Amazon S3 storage for pennies a month.
Some domain registrars toss in some low-spec shared hosting with the registration of a domain name, and those usually include a MySQL database and PHP.
A Google search for "free web host php no ads" returned a small number of not-sleazy-looking companies; but trust - both in terms of my data and their stability - would be enough of a factor for me that I'd look at something less elegant (like hosting from home) before using anyone else.