Is it possible to develop an iOS application for my own use without having to pay 99$/year?
Quite simply, if you jailbreak your iOS device you can likely skip the code expiration checks. We have lots of questions on doing this tethered, so this might not even be a complete duplicate. I’ll leave how to jailbreak for another post since that is a moving target based on your hardware and time of year, release details.
For those that can’t spend the time to reverse engineer this, the $100 is a fair trade off, but for science and for some, I can understand why paying Apple to provide all the tools and code seems attractive if you can keep your test devices secured as designed.
The best way to develop without paying Apple is to use your own tools, and not Xcode. Python, shell, web apps, all sorts of interpreted code and widget apps exist now, as do cloud based solutions that compile down to native code and avoid paying Apple directly.
Your “I need a signed profile” request is the technical term for jailbreaking unless you intend to obtain Apple’s signing keys or rent / borrow someone else’s credentials who has payed to develop.