Eclipse (eclipse.org) is no-install, just drop it into a directory and run it.
Java is a reasonably widespread run-time environment, though as a language, it may not fit the bill for "tinkering." Eclipse supports other languages, too.
If you're looking for a lightweight web container, try Jetty. No installation required, and you can run your own J2EE application (again, if that's "tinkering").
But yes, on your own dime is probably good advice. Look for ways to improve your value to the company. Start with the traditional: learn to do your boss's job (with her/his knowledge, of course).