KeytoSchool.com is a non-profit that runs one of the largest networks of free Moodle hosting in the world. Though it is not an official Moodle partner, from experience I can say that K2S is both responsive and responsible with their hosting and contributions. They provide several domains on which your subdomain installation of Moodle can exist (mine is part of Moodlehub.com) and provide you with a complete installation (plus some).
Along with full administrative control over the system and FTP access and the option to utilize Mahara (not to mention the ability to sell courses freely on the hosted solution, for no fee) you’re also given access to over 80 third-party Moodle plug-ins, blocks and filters. Click the picture below to look at a full list of the modules, blocks and filters that have been included. In addition you’ll also be able to choose from an extensive theme library (and you can even upload your own custom theme). The support forums at K2S are busy and helpful and the founder, Jai Gupta, is courteous and active on Moodle.org and via email.
In a brief correspondence Jai mentioned that they were adding 25 more plug-ins this month alone (possibly including the addition of a mobile Moodle app).
To register visit http://www.keytoschool.com/.
Disclosure: Moodlenews has no affiliation with K2S, Jai Gupta or any other K2S employee apart from utilizing the free hosting service.