MySQL is free software. All software needed to install WordPress is free - Linux, Apache, PHP and MySQL.
If you need a simple website with very low expected traffic, you can use Platform as a Service (PaaS) like AppFog, Red Hat OpenShift like PaaS. They provide a free quota enough to have 200 pageviews per day.
https://openshift.redhat.com/app/
Search on Google web search with How to install WordPress on AppFog (or Red Hat Open Shift) to get guides and tutorial for more help.
It is not possible to tell what your present webhost can do for you. It depends on the hosting plan you are using, thereby the Service Level Agreement (SLA) of the package.