I'm doing a fairly complex one, Serverboards[1]. It is a monitoring, administration and automation framework. And so far without Elixir it would have taken me four times more to develop the backend.

Anyway it does not use Phoenix (the most beloved web framework), but a simple Web server based on Cowboy and JSON-RPC via websockets, so it may not be a canonical example.

[1]. https://github.com/serverboards/serverboards/