Amazon EC2 On-Demand Workers for Short Tasks
There's another option that you may not be aware of. I actually just stumbled upon it: http://multyvac.com
I have no experience using it (so I can't vouch for it), but it looks like the first solution I've seen that actually offers true "utility computing". It began with just Python but now supports any language.
I wish there was some service where I could pay for a highly-available on-demand hardware on a minute to minute basis rather than hourly.
That service is AWS Lambda, which wasn't available when you asked the question:
Lambda runs your code on high-availability compute infrastructure and performs all the administration of the compute resources, including server and operating system maintenance, capacity provisioning and automatic scaling
You are charged based on the number of requests for your functions and the time your code executes
Duration is calculated from the time your code begins executing until it returns or otherwise terminates, rounded up to the nearest 100ms.
The Lambda free tier includes 1M free requests per month and 400,000 GB-seconds of compute time per month.
You can also wrap a Lambda function with an HTTP endpoint, possibly removing this layer from your application:
You can invoke a Lambda function over HTTPS by defining a custom RESTful API using Amazon API Gateway. This gives you an endpoint for your function which can respond to REST calls like GET, PUT and POST. Read more about using AWS Lambda with Amazon API Gateway.
Caveat: Lambda currently supports only JavaScript, Java, and Python, so I'm not sure how you would get R to work. You may need to host R in one of these runtimes.