Amazon does offer a GPU instance, not free by any means.  I did some rough calculations awhile back, LTCwould need to be trading around $25 to show a profit, and BTC would need to be around $500.  (this was about a year ago, I'm sure things have changed since)

I use the AWS EC2 platform for all my clients' websites, and yes, it is free for super low traffic/low power instances.  I tell new clients to create their own account, then give me access via the IAM service.  Works very nicely, and I highly recommend it.