The --format option of inspect comes to the rescue.

Modern Docker client syntax is:

docker inspect -f '{{range .NetworkSettings.Networks}}{{.IPAddress}}{{end}}' container_name_or_id

Old Docker client syntax is:

docker inspect --format '{{ .NetworkSettings.IPAddress }}' container_name_or_id

These commands will return the Docker container's IP address.

As mentioned in the comments: if you are on Windows, use double quotes " instead of single quotes ' around the curly braces.