I was just working on a set of separate Joomla installations for a client today when I realised that I really needed to be able to run scripts against the different installations.
I was trying to install three different Mambots (one of Joomla's three different types of extensions) in about 8 installations of Joomla - each with different database configurations and paths, and having started out with a Bash script to merely copy the plugin files into place, I realised that because automating the whole operation would involve reading a configuration file in PHP syntax and performing some queries in MySQL with it, coding this would probably take longer than installing the plugins manually.
There are not very many web apps which have any kind of scriptable API. In fact, I only really know of Mailman, which is only partly a web application. But it's a feature I've used frequently in Mailman - there is a script
bin/withlist which acquires locks and opens the list, allows you to modify the list as a Python object, and saves it on exit. Mailman provides a few CLI tools too which can be used in scripting but which are really only trivial examples of the power of the scriptable API.
When I began writing Mailhammer, my own announcement-only mailing list software, I took this scriptability even futher based on my positive experience with Mailman's scriptable API. All of the working parts are implemented in Python, and the PHP is just an HTML wrapper which opens and talks to a CLI Python script over pipes. This means that the PHP is kept extremely simple, and the Python core is a very clean and simple API, and that the CLI can do everything reliably. It's a cleanly divided implementation of an n-tier architecture. In fact in practice, I only use the web interface for viewing the data already in the database. Consequently, that interface isn't very powerful - yet!
Python is well-suite for scriptable APIs - its interactive interpreter and neat object model mean that it's easy to perform arbitrary operations interactively on complex, persistent data structures. In PHP web applications it might be more feasible to build an XML-RPC interface of some kind and provide a command-line client.
I don't think that scriptability is considered as even a potential feature for almost any web application I've tried; their operation is tied inextricably to their unique interfaces.
For anybody developing a new web application please ask yourself this: will administrators using your software want to be locked in to your pretty and easy-to-use interface, or will they end up cursing you for failing to provide them with power beyond what HTML can provide?