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So for example: turn-based game servers or even small scale real-time servers should be possible. Chat servers, chatbots. All kinds of automation/administration tools are common usage of the language. For me, Perl keeps an eye my system resource usage and prints the status of it into my window manager. All my non-trivial scripts are written in it, like this one which handles monitors setup: https://github.com/bbrtj/config/blob/system-independent/bin/mon
Perl is Turing complete, so, in theory, anything you can do with any other language, you can also do with Perl. I use Perl a lot in my primary github repo: https://github.com/barrycarter/bcapps but most of that stuff is old, and I've pretty much switched to Python and other tools for the newer stuff
An good way to get started is to go to https://data.gov/ (other countries have similar open data initiatives), look for the datasets that sound like you could be interested in, figure out an interesting question for which the answer might exist in the dataset, and then you have your project scoped.
Really, whatever you like. Finding an idea is often (not always) the tricky part. Here's something small in Perl I've been working on lately, for example: https://gitlab.com/eythian/exhibit-explorer
Here's an example web chat using 30 lines of perl and 50 lines of javascript plus a little html and css.