Some examples of small interesting distributed systems that occur even with a traditional stack:
With multiple visitors on a web app on top of a relational database, you have a small distributed system if they can somehow interact. Even something as simple as a web shop can leave you with an interesting question like what to do when multiple people ordered the last item in stock.
Even just a page with a little bit of ajax going on can give you an interesting small distributed system where a single visitor can interact with themselves.
Or say you need to sync up information between two systems, e.g. export information from the ERP system to a web app or vice versa. That's a small distributed system - there are lots of pitfalls in state transfers.