I'm amazed no one knew of one. Off the top of my head, services which let you run docker containers without dealing with any servers yourself:

1. AWS Fargate

2. Joyent Triton (which was as simple as DOCKER_HOST=<joyent-ip> docker run back in 2015)

3. hyper.sh (available since 2016 I think?)

4. OpenFaaS cloud (though you have to go through a couple more steps, you can bring a custom docker image as your 'function').

5. A metric ton of hosted kubernetes offerings