Did Magento used to offer a hosted solution that included updates or did I imagine it? If not, does any company offer a hosted solution that includes updates?
Magento used to offer a SaaS solution called "Magento Go" but it was shuttered back in July 2014.
AFAIK the hosted solution wasn't well received as you are very restricted on what you can do with the Magento Go store (I believe you can't even see the files/codes) so customizations were limited and difficult.
I highly doubt that any company will be able to offer a hosted solution for Magento that also allows you to customize them but I do believe that there may be a few companies who are offering a dumbed down hosted solution based on Magento but with restrictive customizations.
If you want to customize your Magento store, use your own theme & design, have access to the files/database/codes, you would be better off engaging a web developer to manage the updates for your Magento store.
Magento Go was the platform I was thinking about.
I was just wondering as I have been asked to review a Magento based site by a new client and it hasn't been updated for a while so was just getting back into the loop with it.