Are there any Hosting companies, that offer Geoserver?

If you search with Google for "GeoServer hosting" the very first result offers hosting:

http://www.acugis.com/geoserver-hosting.htm

You might also try:

Webfaction

They seem to be a hybrid between your standard shared server and a hosted server. You have the ability to install pretty much any software that you want. In addition, they will support both the standard LAMP stack, as well as a PostgreSQL backend.

Their pricing is slanted toward a yearly plan, however a monthly plan is only $9.50.

Please note that I am not benefiting financially from this, it is simply my experience with a little research.

Saw a tutorial on running Geoserver on Openshift. You might want to consider it, too.

OpenShift is Red Hat's free, auto-scaling Platform as a Service (PaaS) for applications. As an application platform in the cloud, OpenShift manages the stack so you can focus on your code.

As an added bonus, you can also install the PostGIS spatial database just by adding the Postgresql cartridge. Oh and did I mention that they have a free tier? Good luck :)