Does any CI service let you build Docker containers and use the docker build cache?
CircleCI 1.0
CircleCI lets you build Docker containers as part of your build: https://circleci.com/integrations/docker
You are however right that this doesn't cache the Docker images - it's a bit tricky on our stack to do that right, but we are planning on addressing that soon.
In the meantime, a number of customers have cached it using CircleCI's built-in (non-Docker) caching, by using docker save/docker load: https://circleci.com/docs/docker#caching-docker-layers. It's not ideal, but it should tide you over until we're able to support this a lot better.
Updated for CircleCI 2.0
CircleCI 2.0 is Docker-centric. You can build Docker images, utilize layer caching, use Docker Compose, and even use official and custom Docker images as your build environment.
I'm currently using CircleCI, see http://tschottdorf.github.io/cockroach-docker-circleci-continuous-integration for a good guide to use the docker cache
shippable is also worth checking out, as their entire infrastructure is based around docker and orchestrating multiple docker containers with shippable formations.
Codeship now supports storing the Docker image cache and retreiving this in order to speed up your builds.