I am in a similar situation to you. I also work in a company (distribution and corporate side of after sales parts for an auto manufacturer) that does not allow installation of non-approved software, and while I agree with some of the other posters regarding getting into possible firing territory by going against company policy, sometimes the experience can still be useful to help you move up or around in your company. Try to associate your tinkering with something already related to your job, or to help others in your department. Depending on the size of your company, strictness of management and company policy, you may wind up using a hobby to make lives easier for you and those around you. Or you may get canned. You don't say what kind of work you normally do or what sector of industry you work in, but you have several options. Here are some of the options I have looked at and used.
Another option, of you feel braver is to go the PortableApp route, like you mentioned. You can find portable versions of some scripting languages, such a s Python, which are workable. The downside is that if you decide you need a specific library that doesn't respect being shoehorned into being "portable" (in the sense of being able to run it from an external flash drive or hard drive without leaving traces on the host computer) it could lead to possible discovery by your IT group, depending on how invasive they are in their tracking.
Along in this portable app group, one item I might suggest trying is a scripting language called Rebol. (rebol.com). It's multiplatform interpreter and GUI library in a single file that has some interesting features. Depending on how you run it, it may put a couple of folders in the folder it exists in, but other than that, I think it's pretty "quiet".
Outside of that, some of the posters above have some neat ideas about either remote access to your home computer or utilizing online programming environments. I may even look into these for myself, as the may be feasible depending on what is currently not off limits thought the proxy at work.
By going with the "using what's already available to you" route, whether it's a web browser or an office suite, you may be better suited to present some of your projects and ideas to coworkers and management, since by using those, you technically may not be violating your company's do-not-install rule. It doesn't mean it's bulletproof as they could view creation of new scripts and projects as a violation of the rule, depending on how strict they are. You're the one that has to use your judgement and figure out what you think they will or will not allow, and whether or not it puts you in danger of getting fired.