My spelunking among the public clouds has lead me to the Google Cloud Platform, where I spied with my little eye the Node.js instance and it's $4.49 estimated monthly cost. The cost is not free, but I think it might be the lowest monthly cost for hosting River4 on a cloud server. Probably the only lower costs are either the "free" node on Heroku, with all its constraints, or the one time purchase of a server at CloudAtCost.
I launched the Node.js instance on GCP and found that it is Debian with node.js pre-installed. To install River4, one can skip the Install node.js portion of the Installing River4 on Ubuntu instructions and start with Install forever, then Install git and finally Install river4, before moving on to the River4 howto, starting at step 4.
You will need to open port 1337 in order to access the River4 site, it appears that something (perhaps node.js itself) is using port 80. I haven't done much research into disabling whatever is using port 80. In the Google Developers Console, click Networking, Firewall rules. Click New firewall rule, and enter the fields as follows:
Name: allow-1337
Description: Allow port 1337 traffic to access the River4 home page (this is optional, you could leave it blank)
Network: default
Source filter: Allow from any source (0.0.0.0/0)
Allowed protocols and ports: tcp:1337
Then click Create.
To find the public IP address for your site, on the Google Developers Console click Compute, Compute Engine, VM instances. You will see your instance name listed below a graph showing CPU utilization, and to the right is the External IP address. To access the River4 Home page enter http://[External IP address]:1337 in the address bar of your web browser.
The Google Developers Console provides a button for SSH access to your instance, which opens an SSH session in a browser window.
The GCP Node.js instance is based on the Google Compute Engine f-1 micro instance that can cost as little as $0.0056 per hour per month if you run the instance for a full month. Again, for comparison, the AWS t2.micro instance costs $0.013 per hour, so you can see the compute part of the cost is much lower. I expect there will be additional I/O charges add to the $4.49 estimated monthly price.
I provisioned an instance of Ubuntu 14.04 in Azure to host my River4 Docker container, and it has been running for just over 20 hours with no problems. I am not sure how long I will keep this instance running, I may take it down at any time. Update: I have shutdown the instance and removed the link to it.
There are a couple of things I like about Azure over Amazon. One is that they provide a relatively friendly DNS name for your site. While one can access the River4 home page via a public IP, I think DNS names are much more user friendly and Azure certainly makes that much easier. (Amazon will provide a public DNS but they tend to be really long, and in my opinion, un-friendly.)
The other thing I like about Azure is that they provide a lot of nice, easy to use monitoring tools. In fact, I think the user interface of the Azure portal is much more friendly than Amazon. If you are at all interested in monitoring your River4 instance, I think Azure makes that much easier.
On demand AWS Pricing for a t2.micro instance, which is their smallest, is $0.013 per hour. If you go with a reserved instance for one year, you can get the price down to $0.009 per hour. Azure's on demand pricing for a basic instance, which is their smallest, $0.018 per hour. Azure billing rounds up to the nearest minute, while AWS rounds up to the nearest hour.