It is not only OpenStreetMap insiders who are interested in seeing how a city map or even a whole country's coverage developed over time. Our Geofabrik Labs server has a simple application where everyone can create their own animations.
But while that application only shows the nodes, i.e. the building blocks of OSM data, we provide a full rendering for a few selected areas:
Creating animations like that is not easy because OpenStreetMap is a fast-changing project. Historic data is available but not in the same formats that are in use at OpenStreetMap today, so old versions of software packages have to be used to process old data.
We can create other images like these in any size, style, and format, of any region. We offer this as a paid service to the general public, and free of charge (time permitting) to the OSM community.