I see that Dreamhost is doing something like this but i am not clear how much extra it would cost me. Any other hosts?
It doesn't cost anything extra on DH, and it shouldn't cost anything extra on any other web hosts. All the Gmail integration does is let you choose Gmail as your email provider when you set up a domain. When you do this, DH simply creates the MX records needed to direct all emails for that domain to Gmail's servers. The Google Hosted integration is similar, except now you're actually using Google to host your domains rather than DH's servers. For Google Apps, DH simply creates the DNS records to point docs.example.com, calendar.example.com, sites.example.com, and start.example.com to the respective Google apps.
And for all of these Google services, the DH integration will present you with a link that lets you skip a few steps of the setup process for enabling Google Apps on your domain. But you still need to create a Google Account on your own and manage those apps from Google's control panels.
This is all stuff that can be done fairly easily manually without the integration on any decent web host, but DH has chosen to add the integration into their control panel to make it a little easier since so many people use these services.
They've also added Google PageSpeed and CloudFlare integration, which is more complicated to do on your own (and PageSpeed probably isn't possible on most other shared hosts since it requires changing Apache configurations).
Anyway, as I said earlier, I would not expect most hosts to charge extra for such integration. Either they'll have it built into their cPanel to save you a few minutes of time and offer it for free (which also saves them on customer support requests too), or they won't have it at all. Having users enable these services on their domain requires no extra resources on the web host's part, and in fact offloading email hosting to Gmail saves them bandwidth and server capacity, likewise with Google Hosted. Even PageSpeed is a plus for the web host as it reduces bandwidth usage.