NoSQL: On a shared server, what are the alternatives to using SQL?
My answer to NoSQL: On a shared server, what are the alternatives to using SQL? on Quora
You could probably run Redis on a shared server—it doesn’t need to be installed as root, but it does require a process to run all the time which shared hosts may not allow.
There are a bunch of companies out there that offer hosted NoSQL as a service—https://mongohq.com/ for example. You could talk to those from shared hosting.
Your best bet though would be to upgrade from shared hosting to something more flexible. The bar for building an interesting web app has gone up, and you can get a basic VPS for $10/month—or use a platform like Heroku, ep.io or dotCloud if you don’t want to have to manage your own server.