No we do not.
There's no other way to put it: we're complete control freaks. Our policy is simple: if we don't have a remote-controlled explosive collar welded in place around your neck, we don't give you root on our network. Since there are currently no commercially-available remote-controlled explosive collars licensed for public use, we don't give you root on our network.
This is very good from a security and reliability standpoint, but it does rule out offering these sorts of services.
However, our hosting services have gotten more and more powerful over the years, to the point where you can do pretty much anything on our service that you can do with a VPS, dedicated server, or colo, as long as it's web-based. If you're under the impression you need a VPS or dedicated server to run your web site, try asking about it in our forums, because there might be a better way.
The notable exceptions to this are due to legal restrictions rather than technical ones. For example, handling credit card numbers directly on your site. (We recommend using a third-party service run by security experts instead. We use Stripe.) Similarly, if you are handling regulated health information, there really is no alternative to a provider that can offer HIPAA-compliant hosted services (and, regrettably, the massive associated expense).